Nonlinear Spectral Problems for Degenerate Elliptic Operators

CHAPTER

6

Nonlinear Spectral Problems for Degenerate Elliptic Operators Peter Tak~i~ Fachbereich Mathematik, Universitiit Rostock, D-18055 Rostock, Germany E-mail: takac@ hades.math, uni-rostock.de

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. A priori regularity results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. M a x i m u m and comparison principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The first eigenvalue X1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Convexity on the cone of positive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The inequality of Dfaz and Saa . ~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The first eigenfunction 991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Subcritical spectral problems ()~ < )~1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

389 392 392 393 396 397 398 400 401 402

4.1. Existence and uniqueness for )~ < )~1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Nonexistence for )~ = )~1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. A n t i - m a x i m u m principle for )~ > )~1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Linearization about the first eigenfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Linearization and quadratization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The weighted Sobolev space 79~01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. A compact embedding with a weight for p > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Simplicity of the first eigenvalue for the linearization . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Another compact embedding for 1 < p < 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. A few geometric inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. An improved Poincar6 inequality for p > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

403 406 407 409 410 413 414 417 422 423 427

6.1. Statement and proof of Poincar6's inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Fredholm alternative at X 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

427 433

6.3. Application to the embedding Wo'P ~ L p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. A saddle point method for p < 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Simple saddle point geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. A Palais-Smale condition

435 436 437

.......................................

438

7.3. Fredholm alternative at )~1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

440

H A N D B O O K OF D I F F E R E N T I A L EQUATIONS Stationary Partial Differential Equations, volume 1 Edited by M. Chipot and P. Quittner 9 2004 Elsevier B.V. All rights reserved 385

P Tak6E

386

8. Asymptotic behavior of large solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. An approximation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Convergence of approximate solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. First-order estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Second-order estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. A priori bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6. Nonexistence for ~. -- )~1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. A variational approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. A minimax method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Asymptotic behavior of the constrained minima . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Asymptotic behavior of jz near 4-oo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Existence of a solution for ~. near )~1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5. Existence of two or three solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. (Un)ordered pairs of sub-/supersolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Existence results using ordered pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Existence results using unordered pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. (Un)ordered sets of solutions for )~ = )~1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Bifurcations and the Fredholm alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1. An abstract global bifurcation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Bifurcations from infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Abstract This work surveys analytical methods and results for nonlinear spectral problems for degenerate elliptic operators of the following type: J ~ (u) = 0, which is the Euler equation for the energy functional

&(u)aeflfnA(x, Vu)dx-Zfs2B(x)lulPdx-fs2F(X--p --p

, u)dx

defined on the Sobolev space wI'P(s or WI,p(~(2). Here, S-2 C R N is a b o u n d e d domain, 1 < p < c~, and ~. 6 Ii~ is the spectral parameter (e.g., a control parameter). The energy density in the first (and second) integral in J)~ (u) is a s s u m e d to be positively p-homogeneous in the variable u E R, whereas the reaction function F ( x , .) is assumed to be asymptotically

p-subhomogeneous, F(x,u)

-+ 0

as lul ~

oc, uniformly for x E S2.

lul p The work begins with the properties of the first (smallest) eigenvalue )~1 of the corresponding nonlinear eigenvalue problem, J ~ (u) = 0 with F -- 0. Then the Euler equation J ~ (u) = 0 is studied for )~ < )~1. Finally, the solvability of this equation (existence, nonexistence and multiplicity of w e a k solutions) is investigated for any ~. near ~ l- E m p l o y e d are variational methods (also with constraint), monotonicity methods (pairs of sub- and supersolutions) and asymptotic bifurcation methods (from infinity). A n u m b e r of very recent results on the F r e d h o l m alternative for the (quasilinear) p - L a p l a c i a n A p on W 0' p (S2) is surveyed, most of t h e m with complete proofs.

441 442 445 446 451 457 460 461 462 463 465 468 470 475 476 478 479 482 483 485 487 487

Nonlinear spectral problems

Keywords: Anti-maximum principle, Bifurcation from infinity, Degenerate or singular quasilinear Dirichlet problem, First eigenvalue, Fredholm alternative, Global minimizer, Improved Poincar6 inequality, Minimax principle, Nonlinear eigenvalue problem, p-Laplacian, Saddle point, Sub- and supersolutions MSC: Primary 35P30, 47J10; secondary 35J20, 49J35

387

389

Nonlinear spectralproblems 1. Introduction

The main purpose of this survey is to review some of the most recent developments in the spectral theory of quasilinear elliptic operators of second order and their immediate consequences on solvability of quasilinear elliptic partial differential equations with a spectral parameter. An important class of such equations is represented by the Euler equations for the critical points of the energy functional

J'k (U) def _1 fs2 A (x , Vu)

dx

P

)~ fse B(x)lulP dx -- fse F(x, u) dx P

(1.1)

defined for every function u'X2 --> JR from the Sobolev space WI'p (~2) o r W I'p (~'2), where s is a bounded domain in JRN (N >~ 1), 1 < p < oo, and ~ E JR is the spectral parameter. In applications to engineering problems, )~ can be viewed as a control parameter. The most typical restriction we impose on ,.Tz(u) will be the positive p-homogeneity of the integrands (energy densities) in the first two integrals in (1.1) with respect to the variable u -- u(x) E JR. This means that also in the first integral we require

A(x, t~) = ItlPA(x, ~)

(1.2)

for all t E JR

and for all (x, ~) E f2 x JRN As a consequence, we obtain A (x 0) -- 0 and 0A (X, 0) - - 0. 9

,

The function A(x, .) :JRN __~ JR is assumed to be strictly convex and coercive for every x E I2. Furthermore, we assume that the weight function B :~2 -+ JR is in L~ such that B >~ 0 and B ~ 0 in ~2. Finally, the methods presented in this work apply only to asymptotically p-subhomogeneous integrands in the last integral of J z (u), that is, to

F(x, u)/lul p ~ 0

as lul ~ ~ , uniformly for x ~ S2.

(1.3)

A canonical example of the energy functional (1.1) that we use in a good part of this work is given by & (u) = -

Igul p dx - -

P

P

lul p dx -

F(x, u) dx

(1.4)

on W1' p (I-2), where

F (x , u)

def/o/u -f(x, Jo

t)

dt

for x E s and u ~ R.

The function f : I2 x R --~ R may take, for example, one of the following four forms:

f (x, u) --

g(x); c(x) arctanu 4- g(x); c(x)lulq-2u + g(x);

c(x)lul q-l -t- g(x)

(1.5)

P Tak6(

390

for (x, u) 6 s2 • R, where 1 < q < p, and c, g 6 L ~ ( I 2 ) are given functions which are not both identically vanishing. The corresponding Euler equation for the critical points of the functional ,.Tz defined in (1.4) reads as follows: - - A p u -- ~lulP-2u -4- f (x, u(x))

in s'2;

u =0

on 012,

(1.6)

where f ( x , u) = (OF/Ou)(x, u). Here, Ap stands for the Dirichlet p-Laplacian defined by

Apu def div(lVulP_ZVu)" The first alternative in (1.5), in which f ( x , u) -- f ( x ) = g(x) is independent from the state variable u ~ R for each x ~ S-2, appears to be a typical example suitable for presenting all our basic ideas: Here we develop appropriate analytic tools that can be applied to treat also the three remaining alternatives for f (x, u) without much change. Thus, we will focus our attention mostly on the solvability of the Dirichlet boundary value problem

--mpU -- )~]ulP-2u -q- f (x)

in S2;

u= 0

on 0S2.

(1.7)

Since )~ 6 R is a spectral parameter taking values near the first (smallest) eigenvalue )~1 o f - A p, which is given by (see Section 3)

)~1 = inf, fs2 ]VulP dx" u ~ w~

(s2) with f n ]u]Pdx = 1},

(1.8)

one may regard (1.7) as a problem whose solvability (i.e., existence, nonexistence and multiplicity of weak solutions in W1' p (~2)) should be described by some kind of a nonlinear version of the Fredholm alternative; cf. [37], Chapter II. To investigate the critical points of the functional ,7~, first we need to realize that Jx is coercive on the Sobolev space V whenever )~ < )~1; if V = W~' p (S2) then )~1 > 0, whereas if V = WI'p(ff2) then Xl = 0. Hence, the existence of a critical point, that is a global minimizer for Jx, follows by a standard minimization argument ([53], Theorem 1.2, p. 4). Furthermore, our strict convexity hypothesis on A(x, .) guarantees that the first integral in (1.1) is strictly convex on any linear subspace of W 1,p (S-2) not containing the constant functions. The second integral in (1.1) is obviously convex on L p ($2) and strictly convex on the linear subspace of all constant functions. Finally, if the function F (x, .) :R --+ R in the third integral in (1.1) happens to be convex for each x ~ 12, then fix turns out to be not only coercive (for)~ < )~1) but also strictly convex on V whenever ~. ~< 0. Another wellknown result ([53], pp. 58-60) then guarantees that ,,7~ possesses precisely one critical point, namely, the global minimizer. This is as far as one can get by applying the "general theory" to the functional Jx. If V = WJ 'p (S2), p ~ 2, and 0 < ;k < )~1, the critical points of J~ are not unique, in general: Besides a global minimizer there might also be a saddle point; see [29], Example 2, p. 148, for 1 < p < 2, and [49], Eq. (5.26), p. 12, for 2 < p < oo, where such examples with the function F(x, u) = f ( x ) u are constructed in an open interval S2 C R 1. However, if f ~> 0 and f ~ 0 in 1-2, uniqueness still holds, by a result due to Dfaz and Saa [17] and generalized later by Takfi6, Tello and Ulm [60]. The case )~ < )~l is treated in Section 4.

Nonlinear spectral problems

391

For )~ - ~.1, the coercivity of ,fz~ and consequently the existence of a global minimizer for JZl are lost, in general; see [48], Theorem 1.2, p. 390. Thus, one of the aims of our presentation will be to provide reasonable necessary and/or sufficient conditions on the function F(x, u) such that the functional ,Y'zl have a critical point. In fact, we will obtain additional information on the "geometry" of the functional J z for any )~ near )~1 [21,59]. Finally, we apply topological bifurcation methods to obtain continua of pairs 0~, uz) in 1R • V consisting of a parameter value ~. (near)~ l ) and a critical point u~ for J)~. A standard tool in a number of variational methods is the Palais-Smale condition (at some critical level). Let us now consider only the case V = wI'P(J'2). In [48], Theorem 1.2(ii), p. 390, it is shown that for the functional (1.4) with p > 2 and F(x, u) -f ( x ) u , the Palais-Smale condition fails to hold at the zero level. Therefore, in order to obtain a priori bounds for the critical points of J z for ,I. near )~l, we simply admit possible, a priori large critical and "almost critical" points of J z and then determine their precise asymptotic behavior as )~ approaches )~1. Our method is based on the following well-known fact: )~1 is a simple eigenvalue of the positive Dirichlet p-Laplacian - A p with the associated eigenfunction q)l normalized by q)l > 0 in S-2 and II~0111t~(~) = 1, by a result due to [2], Th6orbme 1, p. 727, and later generalized in [46], Theorem 1.3, p. 157. The corresponding result remains valid also for the first two terms of the more general functional (1.1) on W; 'p (s as shown in [60], Theorem 2.6, p. 80. Moreover, the eigenvalue )~l is positive and isolated. As a consequence, it is not difficult to show ([27], Proof of Th6orbme 2, p. 732, or [28], Section 6, p. 69) that a possible large critical point uz of J z for )~ near )~l must take the form u~ - t -1 (q91 + vtT), where t 6 IK is a number with Itl > 0 small enough, and

vTt E w;'P(s2) is a function orthogonal to ~01 in L2(S-2) with the norm

IlvVtllw~,p(s~) ~ 0

as Itl --+ 0. This forces )~ -- )~(t) ~ )~1 as well. But we need much stronger results on the rate of decay of both, v~ ~ 0 (in a suitable norm) and )~ - ,kl --+ 0 as Itl ~ 0, which have been established recently in [23], Theorem 4.1, and [57], Propositions 5.2 and 8.3, and [58], Proposition 6.1. These results describe asymptotic bifurcations from infinity of the form uz - t -1 (~ol + vtT) as It[ --+ 0, which are easily transformed to bifurcations from zero where the unknown function v~ in tuz = q)l + v~ has to be investigated as [tl --+ 0. Recalling the positive p-homogeneity in u of the first two terms in the functional J z , we notice immediately that the linearization of (1.6) about the eigenfunction q)l together with the "quadratization" of functional (1.4) about q)l play the key role in determining the asymptotic behavior of v~ as Itl --+ 0. In contrast to related methods for the semilinear case p - 2 ( [35], Chapter 18, or [36]), our linearization and quadratization are exact: They use the (precise) integral versions of the first- and second-order Taylor formulas, respectively, rather than Taylor approximations by linear or quadratic expressions. This method was introduced recently by the author [57] and is presented in Section 5. For the special choice f ( x , u) -- f ( x ) the method yields v t~-/[tl p-2t --+ W T as It[ ~ 0, in WI'2(~Q) if 1 < p < 2 and in a suitable weighted Sobolev space D~0~ (Wo 'p (S2) ~-~ 79~0~) if 2 < p < e~. The limit function V ]- is the unique solution of the corresponding limit equation under the condition that V T is orthogonal to ~01 in L2(s The limit equation is linear with the nonhomogeneous term equal to f (x), so that the classical Fredholm alternative for a selfadjoint linear operator in a Hilbert space applies. In a number of important applications we will often be able to show that the asymptotic behavior of large solutions uz - t -1 (q)l + vtT) as Itl --+ 0

P. Tahi6

392

leads, in fact, to a contradiction; for instance, if fs~ f~01 dx ~ 0. Hence, if this happens, there can be no large solutions to problem (1.6) or, in other words, we get an a priori bound on the set of all critical points of functional (1.4). This is the connection between the nonlinear problem (1.6) and the linear problem for V -r obtained in the limit Itl --+ 0. Even if fs2 f~ol dx = 0, our method is precise enough to exclude large solutions, for instance, if )~ = )~l. This is the main difference between the linear case p = 2 and the nonlinear case p ~ 2; see Section 8. The solvability itself of the spectral problem (1.7) is treated by "easier" methods in Sections 6 and 7 (for)~ = )~1), whereas more difficult and complicated tools are developed in Sections 9, 10 and 11 (for)~ near )~l). To summarize the state-of-the-art work on problem (1.7) up to now, many interesting new results have been obtained for )~ near )~l. The work of Anane and Tsouli [4] is one of the very few dealing with the second eigenvalue ~.2 of - A p . A variational characterization of all eigenvalues of - A p is a challenging open question in space dimension N ) 2 (cf. Dr~ibek and Robinson [26]). In space dimension N = 1, when S2 C R 1 is an open interval, significant progress for ~. = )~k (any eigenvalue, k = 1, 2 . . . . ) has been achieved in the recent work of Mamisevich and Takzi6 [47].

2. Preliminaries 2.1. Notation The closure, interior and boundary of a set S C R N are denoted by S_ int(S) and 0S, def

respectively, and the characteristic function of S by Xs" R N ~ {0, 1}. We write ISIN fRN XS(X) dx if S is also Lebesgue measurable. We set R+ - [0, cx~) and N - {1, 2, 3 . . . . }. We denote by S'2 a bounded domain in R N ( N ) 1). Given an integer k ) 0 and 0 ~< ot ~< 1, we denote by C ~'c~( ~ ) the H61der space of all k-times continuously differentiable functions u" S2 --+ ~ whose all (classical) partial derivatives of order ~< k possess a continuous extension up to the boundary and are ot-Hrlder continuous on S2. The norm Ilullc~,=(~) in C k'~ ( ~ ) is defined in a natural way. As usual, we abbreviate C k ( ~ ) - C k ' ~ The linear subspace of C k ( ~ ) consisting of all C k functions u" $2 --+ R with compact support is denoted by C0~(12); we set C ~ ( I 2 ) -- ["]~-0 C~(~2). Given 1 ~< p ~< ~ , we denote by L P (S2) the Lebesgue space of all (equivalence classes of) Lebesgue measurable functions u ' I 2 --+ R with the norm

def{esssup~lu(x) I I,,,(x>l"

Ilullp-IlullL~(s2) -

< oo <

oo

if 1 ~< p < cxz, i f p = cx~.

Finally, for an integer k >~ 1, we denote by W ~'p (S2) the Sobolev space of all functions u 6 L P (S2) whose all (distributional) partial derivatives of order ~< k also belong to L p ( ~ ) . Again, the norm Ilu IIk,p -- Ilu IIw~,~(s2) in Wk'P(S2) is defined in a natural way. The closure in W k'p (S2) of the set of all C k functions u'S2 --+ R with compact support is denoted by

Wko'p (S-2). We refer to Kufner, John and Fu61"k [44] for details about these and other similar function spaces. All Banach and Hilbert spaces used in this article are real.

Nonlinear spectral problems

393

The positive and negative parts of a real-valued function u are denoted by u + and u - , respectively, where u + -- max{u, 0} and u - - m a x { - u , 0}. If u E W I'p ( ~ ) then also u + E w~'P(I2); see [39], Theorem 7.8, p. 153. More precisely, we have Vu + -- Vu almost everywhere in S-2+ -- {x E I2" u(x) > 0} and Vu + -- 0 almost everywhere in I-2 \ S-2+. The corresponding result holds for u - as well. We work with the standard inner product in L2(S2) defined by (u, v) ae__ffs? uvdx for u, v E L 2 (,(-2). The orthogonal complement in L2(S'2) of a set A / / C L2(S'2) is denoted by , / ~ -L'L2 ,

M -t-'L2 de--f{u E L2(~f2): (u, v) -- 0 forall v ~Ad}. The inner product (., .) in L2(S2) induces a duality between the Lebesgue spaces LP(~) and L P'(~), where 1 ~< p, p' ~< ec with ~1 + y1 _ 1, and between the Sobolev space

Wo 'p (I2) and its dual W-1'P'(S2), as well. Finally, this inner product induces also the canonical duality between the space of test functions D ( ~ ) ~ C~C(S-2) and the space of distributions D ~(I2). We keep the same notation also for the duality between the Cartesian products [LP (S-2)] N and [LP'(S-2)]N.

2.2. A priori regularity results We always assume that the function A of (x, ~) E S2 x I[~N and its partial gradient O~A = (~_7~/ji= 1 0 a] N with respect to ~ ~ ~;~N satisfy the following hypotheses, upon the substitution a(x ~)def 1 def 1 oa , = f O~A (x, ~) where ai -- p O~ i 9 HYPOTHESIS (A). A" S-2 x ~N ~ ll~+ verifies the positive p-homogeneity hypotheOA__ sis (1.2), A E c l ( I 2 x IRN), and its partial gradient O~A'S2 x ]1~u ~ I~ u satisfies pI o~i ai E C 1(~f2 x (R N \ {0})) for all i -- 1, 2 . . . . . N, together with the following ellipticity and growth conditions: There exist some constants y, F E (0, cx~) such that

N

Oai Z ~j(X,~)'~it]j i,j=l

N

Oai

~ ) / " I~l p - 2 . I~[ 2

(2.1)

/". I~[ p-2,

(2.2)

/". I~[ p-1 ,

(2.3)

i,j=l N

Z i,j=l

Oai

for all x E S2, all ~ ~ R N \ {0} and all r/E It{ N .

394

P Takd(

(It is evident that it suffices to require inequalities (2.1)-(2.3) for I~1 = 1 only; the general case ~ 6 R N \ {0} follows from the positive p-homogeneity hypothesis (1.2).) It follows that A (x, 9) is strictly convex and satisfies Y p-1

I~l p ~ A(x, ~) <<.

F

I~1p

p-1

for all ~ 6 ]~N.

(2.4)

Hence, the functional J z on Wo 'p (S-2) is coercive and bounded on bounded sets. Indeed, the inequalities in (2.4) follow from y ~l~l p-1

p ~ A ( x , ~ ) - A ( x , O ) - ~. O~A(x,O) ~<

F

p-1

I~l p

for all (x, ~) 6 S2 x ~;~N. This is, in turn, a direct consequence of Taylor's formula combined with (2.1) and (2.2). Recall that the positive p-homogeneity hypothesis (1.2) forces A (x, 0) = 0 and ~0A(x / . 0 ) = .0 f o r a. l l x 6. S 2 a n. d i - -.1 . 2, N Finally, we assume that F satisfies: HYPOTHESIS (F). F :s

x 1R --+ 1R is given by the integral

F ( x , u ) -- fo b/ f ( x , t ) d t

f o r x ES2 a n d u E R ,

where f :S-2 x R ~ R is a Carath6odory function, i.e., f (., u) :s --+ IR is Lebesgue measurable for each u 6 IR and f (x, .) : R --~ R is continuous for almost every x 6 S2, and there exists a constant C 6 (0, c~) such that

If(x, u) I ~< C(1

+ lul p - l )

for all x ~ S2 and all u ~ N.

(2.5)

Now we are ready to state the main regularity result for a weak solution u E W~ 'p (I-2) of the Dirichlet boundary value problem - d i v ( a ( x , V u ) ) = f ( x , u(x))

in n ;

u= 0

on 012.

(2.6)

We will use this a priori regularity throughout the entire article. PROPOSITION 2.1. Let 1 < p < cx~ and let hypotheses (A) and (F) be satisfied. Assume that u E w;'P (s2) is a weak solution of problem (2.6). Then u ~ C1'/~($2) where/3 E (0, 1) is a constant independent from u. If, in addition, 0 S2 is a compact manifold of class C 1,a for some ot E (0, 1), then ~ ~ (0, ~) can be chosen such that u E C l'~ (~). Moreover, ~ is again independent from u, and IlUllcl,~(~) <. C where C > 0 is some constant depending solely upon S2, A, f , N, p, and the norm IlUllLp0(n) with p,

po--

2p

Np

= N-p

if p < N, if p >~N.

Nonlinear spectral problems

395

Notice that, owing to the Sobolev embedding W l ' p ( s L P~163 we have also Ilullcl,~(~) ~< C', where the constant C ~ depends solely upon s A, f , N, p, and the norm Ilullwl,p(s2). Similarly, one obtains IlUllcl,~(~) ~< C" as well, where the constant C" depends solely upon s A, f , N, p, and the norm Ilullg~(s2). These two consequences of Proposition 2.1 will be used quite often in the sequel. Proposition 2.1 is, in fact, a combination of the following two lemmas, in which we keep our hypotheses and notation from the proposition. L E M M A 2.2. Let g ' s x IR ~ 1R be a Carathdodoryfunction such that g(., s) E L~oc(s f o r every s E R, and the following inequality holds with some constants a > 0 and b ~ 0:

s . g ( x , s) <<.alsl p + blsl Assume that u E W l ' p (s

f o r all s E R and a.e. x E s

satisfies

~ ( a ( x , Vu), Wp)dx -- fs2 g ( x , u(x))dpdx

f o r all 4) E C~C(s

Then u E L ~ (s and there exists a constant c > 0 such that pends solely upon a, b, N , p, and Ilu Ilgp0(re).

Ilu IlL-(n)

~ c, where c de-

This is a special case of a more general result shown in Anane's thesis [3], Th6orbme A.1, p. 96. Although his proof is carried out only for 1 a(x,~)---O~A(x,~)-l~lP-2~,

(x, ~) E s

X]~ N ,

(2.7)

P one can rewrite it directly for our more general case. LEMMA 2.3. Assume that u E W; 'p (s is a weak solution o f problem (2.6) such that u E L~163 Then u E C1'~(s where/3 6 (0, 1) is a constant i n d e p e n d e n t f r o m u. If, in addition, 0s is a compact manifold o f class C i'~ f o r some ~ E (0, 1), then fl ~ (0, or) can be chosen such that u E Cl'~ ( ~ ) . Moreover, fl is, again, independent from u, and Ilullc,,~(~) <<.C' where C ~ > 0 is some constant depending solely upon s A, f , N , p, and the norm Ilu IIL~(S2).

The first statement of this lemma, interior regularity in C l'fl (if2), was established independently by DiBenedetto [ 18], Theorem 2, p. 829, and Tolksdorf [62], Theorem 1, p. 127. The second statement, regularity near the boundary, is due to Lieberman [45], Theorem 1, p. 1203. The constant/3 depends solely upon or, N and p. We keep the meaning of the constants ot and fi throughout the entire article and denote by fl~ an arbitrary but fixed number such that 0 < fl~ < fl < o~ < 1. Last but not least, Lieberman's regularity results have been shown for the Neumann boundary conditions as well.

396

P. T a k d (

While Anane' s proof of Lemma 2.2 is based on the special form of a(x, ~) = 71 0f A (x, ~) with the positively p-homogeneous potential A(x, .) satisfying also hypothesis (1.2), L e m m a 2.3 is valid with any vector field a -- --,-(ai)-N1 9 ~ X R N ~ I~ N satisfying

ai 9 O(I'2 x ]RN) rqC l(l2 x (JRN \ {O}))

(i=l,2,...,N)

together with the ellipticity and growth conditions (2.1)-(2.3).

2.3. Maximum and comparison principles The strong maximum principle for the critical points of the functional ,.7o (i.e., X = 0) will turn out to be essential in proving the simplicity of the first eigenvalue X1. We begin with the weak comparison principle for the weak solutions u, v 9 W~' p (S2) of the following Dirichlet boundary value problems, respectively, - div(a(x, g u ) ) + b(x, u) = f ( x )

in S-2;

u= 0

on 0S2,

(2.8)

- d i v ( a ( x , Vv)) + b(x, v) = g(x)

in S-2;

v= 0

on 0S2.

(2.9)

As a direct consequence, the uniqueness of these solutions follows. We assume that A satisfies Hypotheses (A), and b : $2 • R ~ R is a Carath6odory function that is increasing in the second variable, i.e., u ~< v in R implies b(x, u) <<,b(x, v) for almost every x 9 12, and it satisfies the growth condition (2.5) with b in place of f . LEMMA 2.4. Let f, g 9 L ~ (12) satisfy f ~ g in I2. Then any weak solutions u, v 9 Wo 'p (~2) of problems (2.8) and (2.9), respectively, satisfy u <, v almost everywhere in ~2. This result is shown in [61], Lemma 3.1, p. 800. Its proof is standard: Consider the function w -- (u - v) + def max{u -- v, 0}; hence, w 9 Wo 'p (S2). Subtract the second equation (2.9) from the first one (2.8), multiply the difference by w, and subsequently integrate over $2. The integrals over ~2+ = {x 9 C2: w(x) > 0} force IC2+IN = 0. To obtain the strong maximum principle for a weak solution u 9 Wo 'p (12) of problem (2.8), we strengthen our hypotheses on b as follows:

HYPOTHESIS (b). b : I 2 • IR --+ IR is a Carath6odory function that is increasing in the second variable and satisfies the growth condition

Ib(x, u)[ ~< Clul p-1

for a.e. x 9 s2 and all u 9 ~,

(2.10)

with a constant C 9 (0, c~). Recall that ~2 is said to satisfy an interior sphere condition at a point x0 e O~ if there exists an open ball Br(y) = {x e RN: ix -- Yl < r}, centered at some point y 9 $2 and with radius 0 < r < cx~, such that Br (y) C $2 and xo 9 OBr (y).

397

Nonlinear spectral problems

LEMMA 2.5. Let f ~ L~(S-2) satisfy f ~ 0 and f ~ 0 in Y2. Then any weak solution u E w~'P(I-2) o f p r o b l e m (2.8) verifies u > 0 almost everywhere in ~ . If, in addition, S-2 satisfies an interior sphere condition at a point xo E 0 I-2 and, f o r some e > O, 01-2 f-1 Be (xo) is a manifold o f class C 1 and u E C 1( ~ f-) Be (xo) ) , then the outer normal derivative on 01-2 of u at xo verifies the H o p f maximum principle (Ou/Ov)(xo) < O. This result is due to Tolksdorf [61 ], Propositions 3.2.1 and 3.2.2, p. 801, for a(x, ~) a(~) and to V~izquez [65], Theorem 5, p. 200, for

1 a(x, ~) ~ --O~A(x,~) --I~ iP_2~ , p

(X,~) E if2

X

]/~N.

The proof given in [61 ], p. 802, extends directly to our more general case. REMARK 2.6. One may ask if the strong comparison principle, u < v

inE2

and

Ou Ov

>

Ov Ov

on0E2,

(2.11)

is valid in the setting of L e m m a 2.4, provided f ~ g, OE2 is of class C l,~ for some c~ (0, 1), and S2 satisfies the interior sphere condition at every point of 0S2. We refer the reader to [12,13] for a positive answer to this question in a number of special cases; for example, if b satisfies hypothesis (b) and, in addition, b(x, .) is locally Lipschitz continuous on • \ {0} for almost every x E 12, and

Ob

{ F . lul p-2

o<<.~(x,u)<~ r

if 1 < p < 2, if 2 ~< p < cx~,

(2.12)

holds for almost all (x, u) E 12 >< (0, e0], with some constants F E (0, oo) and e0 > 0, these hypotheses guarantee only u < v near the boundary of E2 together with Ou/Ov > Or~Or on Of2 for any 1 < p < o~ and 0E2 connected ([13], Proposition 2.4, p. 728). For 1 < p <~ 2 they guarantee also (2.11) provided either N = 1 ([13], Theorem 3.1, p. 733) or else N >~2 and u(x) = u(lxl) and v(x) = v(lxl) are radially symmetric in a ball s = BR(O) C I[{ u ([13], Theorem 3.3, p. 737). However, for p > 2, Hypotheses (b) and (2.12) may not be sufficient for (2.11) to be valid throughout the domain S2: A counterexample to (2.11) with u(0) = v(0) is given in [13], Example 4.1, pp. 740-741, where 12 = B1 (0) C ]~N is the unit ball and b(x, u) =~,lulP-Zu with a constant )~ > 0 large enough. Unlike in [61], Proposition 3.3.2, p. 803, and in a number of other articles on this topic, in [12,13] it is not assumed that IVul > 0 or IVvl > 0 throughout S2. In fact, any function u 6 W1' p (S2) n C 1(~) must attain its maximum or minimum in E2 at some point ~ 6 f2; hence, V u (~) - 0.

3. The first eigenvalue )~l Let us consider the energy functional ffz defined by (1.1). We assume that A satisfies Hypotheses (A), and the weight function B satisfies:

P Tak6(

398

HYPOTHESIS (B). B'S2 ~ R+ belongs to L~(Y2) and does not vanish identically (almost everywhere) in S2. We treat the more difficult case of ffz defined on W~ 'p (S2) in detail, leaving the trivial case ,,7z" wl'P(ff2) ~ ]1~ to the reader. The first (smallest) eigenvalue )~1 for the Euler equation corresponding to the energy functional 3"~ on

W~ 'p (~2) is given by

fs2 A(x, Vu)dx" u 6 wI'P(s'2)with fs2 B(x) lulPdx - 1 } .

~, 1 - - i n f ,

(3.1)

Since the Sobolev embedding W01'P(12) ~ LP(S2) is compact by Rellich's theorem, the infimum above is attained and satisfies 0 < )~1 < c~. Furthermore, it is easy to see that if u ~ WI'p (12) is a minimizer for )~1, then so is c~u+ provided u + ~ 0 in s and ot --

-+-(fs2 B(x)(u+)P dx)-l/P" The corresponding claim holds also for u - , with the constant ot fl = • B(x)(u-) p dx) -1/p. Indeed, if both u + ~ 0 in S2, then we have

replaced by

~,1 = fX2 B(u+)P fx2 A(x,

Vu +) dx

fo BlulP fo B(x)(u+)P dx

>i

fs2 B(u-)P fs2 A(x, V u - ) d x fs2 BIuIP fs2 B(x) (u-)P dx

+

{ fs2 B(x)(u+)P + fs2 B(x)(u-) p )

fog(x)lulp

fse B(x)lulP

=~,1. Consequently, both otu + and the Euler equation -div(a(x,

flu- are (nontrivial) minimizers for ~,1 and therefore satisfy

Vu)) -- )~l B(x)lulp-2u

in S2;

u= 0

on 0S2.

(3.2)

We apply the strong maximum principle (Lemma 2.5) to conclude that u + ~ 0 in s forces u > 0 almost everywhere in S2, and analogously for u - . Thus, a minimizer u ~ W~ 'p (I-2) for )~1 is either almost everywhere positive or else almost everywhere negative in S-2. Our next goal is to show that a minimizer u 6 W1' p (S2) for )~1 is unique up to the sign 4-. In other words, we wish to show that the eigenvalue )~1 in problem (3.2) is simple. Notice that the case ,7z : W 1,p (S2) ~ R is trivial due to the fact that )~1 = 0. Hence, the only minimizers for )~1 over W 1,p (S-2) are the constant functions u - + (fs2 B (x) dx) - l / p.

3.1.

Convexity on the cone of positive functions

u w-> fs2 A(x, V u ) d x is strictly convex on w~'P(I-2), by the ellipticity condition (2.1) (hypothesis (A)). Knowing that any eigenfunction u ~ Wo 'p (I-2) associated with the first eigenvalue ~,l, that is to say, any weak solution u E Wo 'p (S-2) to

Notice that the functional

problem (3.2), must be either positive or else negative throughout s

we may replace u

399

Nonlinear spectral problems

by - u if necessary and thus assume u > 0 in I-2. Hence, the minimization constraint in formula (3.1) reads fn B(x)u(x)P dx -- 1. Upon the substitution becomes linear in v, i.e., fn B(x) v(x) dx = 1. It follows that

def

v -- u p,

the constraint

~.1 -- inf, fn A(x, V(vl/P))dx" v 6 (/+ with fs? B(x)v(x)dx --1},

(3.3)

where

~+ de__f{11" ~ ~ (0, 0(3)" Ulip E w;'P(~'2)}. The following property of the functional v ~ step in proving that the eigenvalue X1 is simple.

fn A(x, v ( v l / P ) )

dx on ~r+ is the crucial

DEFINITION 3.1. A f u n c t i o n a l / C ' C ~ IR U {+ec} on a convex cone C C X \ {0} in a vector space X (over the field N) is called ray-strictly convex if, for all v0, Vl E C and 0 6 (0, 1), we have K2((1 -

O)vo + Ovl) <~(1

-

O)l~(vo) + 0K~(Vl),

where the equality may hold only if v0 and v~ are co-linear, i.e., v l = oev0 for some ot

(0, ~ ) . Recall that C is called a convex cone in X \ {0} if C C X \ {0} is convex and satisfies v 6 C =~ otv E C for all oe 6 (0, ec).

LEMMA 3.2. The functional 1~" (/+ ~ 1R, defined by l~(v) de__ff~2 A(x, v ( v l / P ) ) dx for

v ~ % , is ray-strictly convex on (/+. Notice that the statement of the l e m m a includes also the convexity of the set 1)'+. This l e m m a is shown in [60], L e m m a 2.4, p. 79. For the special case A (x, ~) = I~ Ip, (x, ~) S2 x R N, it is due to Dfaz and Saa [ 17]. PROOF OF LEMMA 3.2. Using the positive p - h o m o g e n e i t y hypothesis (1.2), we observe that

A(x, v(vl/p)) = p-Pv A(x,

v-iVy)

for all v 6 l;'+ and a.e. x 6 s

Take v = (1 - O)vo + OVl and replace Vv by ~ = (1 - 0)~0 + 0~l, where v0, Vl E (0, OO) and ~0, ~l 6 R N are arbitrary positive numbers and arbitrary vectors in R N, respectively, and 0 6 (0, 1). Next we rewrite (1 - O)vo ~o O)vo + Or1 (1 - O)vo + Or1 vo ( 1 - O)vo ~o } OVl ~l (1 - 0)~o + 0~1

v

(1 -

V

VO

V

Vl

(1 -

Or1 ~1 O)vo + Ovl Vl

400

P Takd(

Now fix any x 6 S2. The function A(x, .) being strictly convex on ]~N, by the ellipticity condition (2.1) (Hypothesis (A)), we compute

A ( ~x,)

<~ (1

-- O)V~ (~0) x,-v

vo

=(1-O)V~ The equality holds if v a ( x , v - l ~ ) is convex In particular, taking integrating over ~2, we and

l)

x,

1)0

+

OVl A (~1) x,-v

+0~A

/3

Vl

x,~

Vl

.

(3.4)

and only if ~o/vo - - ~l/Vl. Furthermore, the function (v,~) w-~ on (0, oo) • IRu. v0, Vl 6 ~'+ and ~i = V V i (i = O, 1), multiplying (3.4) by v and arrive at (1 -O)vo +Ovl ~ ~'+, proving the convexity of the set 'r

/C((1 - O)vo + Ovl) <~ (1 - O)lC(vo) + 0KS(Vl) where the equality holds if and only if l)olVv0 -- l)llVVl in s equivalent to Vl/vo -- const in I-2. The lemma is proved.

The latter equality is K]

3.2. The inequality of Diaz and Saa An important consequence of the ray-strict convexity of the functional/C" ~'+ --+ R established in Lemma 3.2 is the following ray-strict monotonicity of its subdifferential 3/C" ~'+ --+ Dr(S2) defined formally for each "suitable" v0 6 ~'+ by

OK~(vo)de~ _

div(a(x, V(vl/P)))

(3.5)

V (oP-1) / P

For v0 6 1~'+ we write v0 6 dom(0/C) if and only if (i) ess infK v0 > 0 on every compact set K C s and (ii) the expression in (3.5) belongs to D ~(s Substituting uo -- rio/p above we get

O 1 C ( u ~ ) - - div(a(x, Vu0))

u~ -1

REMARK 3.3. We claim ~'+ A C~163 C dom(0K~). Indeed, if v0 6 ~'+ r C~ its "neighborhood"

N(vo) -- vo -k- BK,~ = {v0 -k- ~b: ~b E BK,3}, where

BK,~ -- {r ~ Col (Y2)" r = 0 in s \ K and

IIr

< s},

then also

401

Nonlinear spectral problems

satisfies N(vo) C 1~'+for any compact set K C 1-2 and for some sufficiently small number -- 6(v0, K) > 0. It is now easy to see that the functional/C has the directional derivative at v0 in every direction ~b 6 C~ (~2) \ {0}. According to (3.5), this derivative is given by (0/C(v0), 4 ~ } - / ; a(x, V(v~/P)) 9 V ( v o ( P - I ~ / P ~ ) dx. Hence, O)U(vo) is in the dual of the Fr6chet space C 1 (S-2) and, in particular, in D'(I-2). The following ray-strict monotonicity is a generalized version of the well-known inequality of Diaz and Saa established in [17] for the special case A ( x , ~) = I~l p, (x, ~) 12 • ~N. Their hypotheses have been weakened by Lindqvist [46]. Here we state this inequality under the hypotheses convenient to us. LEMMA 3.4. Letuo, ul ~ Wol'P(12) be such thatuo > O a n d u l > 0 in f2 a n d b o t h and Ul/UO are in L ~ ( 1 2 ) . Then we have fs~(

div(a(x, Vu0)) + div(a(x,-jV u l ) ) ) (ug - u~)dx ~>0 u~-1

UO/Ul

(3.6)

uf

where the equality holds if and only if vl /vo = const in 12.

Of course, the integral in (3.6) has to be understood as

fs2 [a(x, Vu0). V(u 0 --(Ul/Uo)P--IR1) + a(x, V u , ) . V(ul - ( u o / u l ) P - ' u o ) ] d x ~ O.

(3.7)

The last integral converges absolutely as a Lebesgue integral. Moreover, its integrand is always nonnegative owing to inequality (3.4); it vanishes if and only if ( V u o ) / u o = ( V u l ) / U l . However, if we know that both expressions div(a(x, Vu0)) and div(a(x, Vul)) are, say, in L ~ (S2), then also the integral in (3.6)converges absolutely. The proof of our generalized version of the Dfaz-Saa inequality is analogous to those in [17] and [46]; we leave the details of proving (3.4) =, (3.7) to the reader.

3.3. The first eigenfunction ~ol The inequality of Dfaz and Saa is often used to show that the eigenvalue )~1 in problem (3.2) is simple, cf. [17] and [46]. In order to avoid any smoothness hypothesis on the boundary 012, we prefer to apply Lemma 3.2 directly. COROLLARY 3.5. Let u0, us E W~ 'p (12) be two eigenfunctions associated with the eigenvalue )~1, i.e., let uo and u I be two nontrivial weak solutions to problem (3.2). Then either uo > 0 or else uo < 0 throughout 1"2, and u l/uo =- const in 1-2.

P Tak66

402

This corollary is a nonlinear version of the well-known Krein-Rutman theorem for linear operators. If 0 S-2 is of class C 1'c~ for some ot 6 (0, 1), and S2 satisfies the interior sphere condition at every point of 0S2, then the Hopf maximum principle (Lemma 2.5) applies to (3.2), and so the abstract nonlinear KreYn-Rutman theorem from [55], Theorem 3.5, p. 1763, can be used to derive our corollary9 An alternative proof, using Picone's identity for the p-Laplacian, can be found in [ 1], Theorem 2.1, p. 821. Another abstract version of the nonlinear Krein-Rutman theorem is given in [41 ]. PROOF OF COROLLARY 3.5. Owing to formula (3.1) we observe that both u0 and Ul are minimizers for the functional '7"(0) '-"x, (U) de~ _1 p fs~ A (x, Vu) dx

X1 fs-e B(x)lulP dx P

defined for u ~ W~ 'p (S2). By our reasoning at the beginning of this section, we know that the function ui (i = 0, 1) has definite sign = +1 throughout 12; we may assume ui > 0 in S2 9 Hence, the function l)i--U p satisfies 13i E V+ and q4~ --0 " Now we apply ~")~ 1 Lemma 3.2 to conclude that the functional K~ 1 (v)def

p,j(O)(vl/P): fs2 A(x, v(vl/P))dx-)~l fs-2B(x)vdx,

v~ f'+,

is ray-strictly convex on f'+. Consequently, if Vl/vo ~ const in S2, then for any convex combination v -- (1 - O)vo + Or1 with 0 6 (0, 1) we must have K:x, (v) < (1 - 0)Kk, (v0) + 0Ex, (v,) = 0, a contradiction to formula (3.1). We have proved

U l / U 0 ---- const

in S-2 as desired.

73

REMARK 3.6. According to Corollary 3.5, from now on we denote by ~01 the positive solution to problem (3.2) normalized by the condition II~0111zP= 1. In this way ~01 is determined uniquely. Recall that the strong maximum principle (Lemma 2.5) guarantees ~01 > 0 almost everywhere in 12. Moreover, if the boundary 0 S'2 is of class C 1,~ for some 6 (0, 1), then ~01 6 Cl'/~(~) for some fl E (0, or), by Proposition 2.1. Finally, if 12 satisfies also an interior sphere condition at a point x0 E 012, then (Oq91/Ov)(xo) < 0, by the Hopf maximum principle (Lemma 2.5). We will need these facts throughout the rest of this work.

4. Subcritical spectral problems ()~ < )~l) We are concerned with the critical points of the energy functional ,.Tx on Wl ' p (12) defined by (1.1) for the "subcritical" values of the spectral parameter ~, -cx~ < ~ < ,k 1. We assume

Nonlinear spectral problems

403

that A and B satisfy Hypotheses (A) and (B), respectively, and the function F satisfies Hypothesis (F) together with the decay condition f (x,u) lulP-1

--+ 0

as lul ~ ~ , uniformly for x ~ S2.

(4.1)

This means that f = OF/Ou is assumed to be asymptotically ( p - 1)-subhomogeneous; cf. (1.3). The existence of a critical point uo ~ W~ 'p (f2) for Jz on W l'p (1-2), that is a global minimizer, is a textbook result obtained by a standard minimization argument ([53], Theorem 1.2, p. 4). However, the uniqueness of this critical point depends on the geometry of the functional Jz. If, for instance, )~ <~ 0 and the function u v-+ f (x, u) is decreasing on R for a.e. x ~ ~2, then both functions u w-> -)~lul p and u w-> - F ( x , u) are convex, and therefore the functional J z is strictly convex on W~ 'p (1-2). This shows that the global minimizer is the only critical point for J z ([53], pp. 58-60). The case 0 < ), < )~l is more delicate and so one needs to be more specific in addressing the question of uniqueness of a critical point. Let us consider only the case f (x, u) ~ f (x) independent from u 6 R where f 6 L~(S2). Examples constructed in [29], Example 2, p. 148, for 1 < p < 2 and [49], Eq. (5.26), p. 12, for 2 < p < exz, both in an open interval 1-2 C R1, show that besides a global minimizer there also might be a saddle point for Jz. In these counterexamples to uniqueness, the function f changes sign in the interval $2. The next theorem shows that this is essential.

4.1. Existence and uniqueness for )~ < )~l THEOREM 4.1. Let --c~ < )~ < ~1. Assume that A and B satisfy Hypotheses (A) and (B), respectively, and the function F satisfies Hypothesis (F) together with the decay condition (4.1). In addition, let f ~ 0 in ~2 • R and assume that the function u w-> f (x, u ) / u p-1 is decreasing on (0, c~) for a.e. x E ~ and strictly decreasing for all x ~ $2' from a set 12' C 12 of positive Lebesgue measure. Then the Dirichlet boundary value problem - div(a(x, Vu)) -- X B(x)[uip-2u u--O

+ f(x, u(x))

in $2, on 012,

(4.2)

1

possesses a weak solution u E Wo 'p ($2). Moreover, if f ( . , 0) ~ 0 in 1-2 then u > 0 in 12, and this solution is unique. On the other hand, if f (., O) =- 0 in 1-2 then, besides the trivial solution ~ 0 in 1-2, problem (4.2) possesses at most one nontrivial solution u; it satisfies u > 0 in S-2 again. Finally, the nontrivial solution (ifit exists) is the global minimizer for ,7~ over W 1' p (12). For the special case A(x, ~) = I~1p, (x, ~) ~ S2 x •N, this theorem was first obtained by D/az and Saa [ 17], Thtorbme 1 et 2, p. 521. The method of proof we present below is taken from [29], Appendix and [60], Proof of Theorem 2.5.

P. Takd(

404

PROOF OF THEOREM 4.1. We begin with a trivial observation: The positive p-homogeneity hypothesis (1.2) implies A (x, ~) = ~ 9a(x, ~) for (x, ~) 6 12 • N N, where a(x, ~) def = ~1 0f A (x, ~). This is an easy computation,

A(x, ~) - A(x, O) -- p fo 1 ~. a(x, t~) dt --p

(f0' ) tp-ldt

~ 9a(x, ~) = ~ 9a(x, ~).

Let u ~_ W l'p (X?) be any critical point for J)~ on lution of problem (4.2). We claim that u ~> 0 in s

Wo 'p (s

i.e., let u be a weak soIndeed, we can multiply (4.2) by

u - 6 W0L p (~2) and then integrate by parts over X2, thus arriving at

fs? a(x, Vu) " Vu- dx = X fs? B(x)]ulP-2uu- dx -+-fs? f (x, u(x)) u- dx. (Recall u - = m a x { - u , 0}.) Employing A (x, ~) = ~ 9a(x, ~) and f ~> 0 in ~2 x R, we get

fs2 A(x, V u - ) d x <~)~fs2 B(x)(u-)P dx. Formula (3.1) and )~ < ~,1 then force u - = 0 in 12, i.e., u ~> 0 in s The a priori regularity result in Proposition 2.1 applied to problem (4.2) guarantees u 6 C l ' ~ ( f 2 ) for some/3 6 (0, 1). The strong maximum principle (Lemma 2.5) leaves us with the following two alternatives: Either u > 0 throughout f2, or else both u = 0 and f ( . , 0) -~ 0 in S2. It remains to treat only the former alternative; we have to show that fix

u ~ Wo 'p (~) with u > 0 in 1-2. To this end, let u0 denote a global minimizer for ,.74 on W; 'p (S-2). Hence, u0 ~> 0 in 12 and ,.7~(uo) <~ ff~ (0) - O .

has at most one critical point

Moreover, we have either u0 > 0 in S2, or else u0 --= 0 in 12. Suppose that u 1 6 W;' p (S2) is another critical point for ffz on W0l' p (S2) such that u 1 ~ 0 in S2; hence u 1 > 0 in f2. We will arrive at a contradiction, i.e., we show that u l -= u0 in f2. Set vi -- u/p (i - 0, 1); so v0 E r~+ U {0} and Vl ~ k'+. Next we show that the functional v ~ J z (/)l/p) is strictly convex on f'+ t_J {0}. Indeed, define

j.(0) (U) de__.f_1 fs? A (x , Vu) dx P

)" f ~ B(x)[u[ p dx, P

U G_w;'P(ff2).

L e m m a 3.2 shows that the functional v w-~ J z ~~ (1) l/p) is ray-strictly convex on 'v'+. Furthermore, employing our hypothesis that the function u w-~ f(x, U)/bl p-1 is decreasing on (0, cx~) for a.e. x 6 S2 and strictly decreasing for all x 6 S21 from a set S2 f C S2 of positive Lebesgue measure, we obtain that the function v ~ - F ( x , v I/p) is convex on

Nonlinear spectral problems

405

R+ = [0, cx~) for a.e. x 6 12 and strictly convex for all x 6 12 ~. We conclude that the functional K~x (v) de__fp j .

(vl/p)

= pj~O)(vl/P)

__ P f~

F(x,

V 1/p) d x ,

l) 6 V+ U {0},

(4.3)

is strictly convex. Now we are ready to show u i = u0 in Y2. Suppose that u l ~- u0 in $2 and consider the function x(O)--/Cz((1-O)vo+Ovl)

forO~
Recall Vi --/,//P (i = O, 1) with v0 E 1)'+ U {0} and Vl E ~r+. The function u0 being a global minimizer for J z on W~'p (1-2), we must have x(O) >~x(O) for all x E [0, 1]. Moreover, x is strictly convex on [0, 1], by the strict convexity of K~z. Elementary analysis results then imply 0 ~< lim 0--+0+

x(0)

- x(0)

0

< lim

x(1)

t-+0+

- x(1 -

t

t)

,

(4.4)

for the one-sided derivatives of a convex function exist and are increasing. Since ul E Cl't~(I2) and ul > 0 in ~2, the subdifferential 0K~z(vl) of/~z at vl exists in 79f($2) and is given by (see Remark 3.3) 0K~Z(Vl)

=

-

~

1

[div(a(x,

V(v '/p)

(x)) 1/p

v1

=-

1

(p_,~/----~J~(v~/p)-"'

"

inS'2,

(4.5)

vI

where J~" W~' p (12) --+ W - l , p'(l-2) stands for the Fr6chet derivative of J z on In the norm of

w~'P(12)

W~'p (n).

we approximate the difference v 0 - vl by a test function 4~

from D(I2) (hence, with compact support) in order to guarantee Vl + t4) >~ 89Vl > 0 for all 0 ~< t ~< tl, where tl ~ (0, 1) is a sufficiently small number. If 4~ is taken close enough to v0 - vl, then the second inequality in (4.4) remains valid also when v0 - vl is replaced by 4~. That inequality shows that we cannot have 0K~Z(Vl) = 0 in DI(S2). Consequently, neither can the equation Jf(vl/p) - 0 hold in W-l,P'(12). We conclude that Ul - - l)I/p is not a critical point for J z , a contradiction to our assumption. We have verified that u0, a global minimizer for J z over W~' p (I2), is the only nontrivial critical point for J z if u0 ~ 0 in 1-2, and the only critical point if u0 ~ 0. This finishes our proof of the theorem. D Even though the two remaining paragraphs of this section deal with two special problems which are critical (X = )~l) and even supercritical (X - X1 > 0 small enough), we have

406

P Tak6g

decided to include them here because these problems can be treated by the same methods that we have used in the proof of Theorem 4.1.

4.2. Nonexistence for )~ =

~,1

Much of the present article is devoted to the question of solvability of the critical spectral problem - div(a(x, Vu)) = X1B(x)lulP-2u -t- f ( x ) u=O

in s

(4.6)

on OS2.

The following nonexistence result complements Theorem 4.1. It will turn out to be extremely useful later in a detailed analysis of certain asymptotic properties of large solutions to problem (4.2) for ,k near )~l (~. > )~l). THEOREM 4.2. Assume that A and B satisfy hypotheses (A) and (B), respectively, and f ~ L~(~2) satisfies 0 <. f ~ 0 in I2. Then problem (4.6) has no weak solution u ~ W~ 'p (S2). For the special case A(x, ~) = I~1p, (x, ~) ~ S2 x ]~N, this theorem is due to Fleckinger et al. [27], Throrbme 1, p. 731, or [28], Theorem 2.3, p. 54. A different proof thereof, based on Picone's identity, is given in [ 1], Theorem 2.4, p. 824. PROOF OF THEOREM 4.2. On the contrary, suppose that uo E W; 'p (1-2) is a weak solution of problem (4.6). One shows u0 6 CI'~(S-2) for some 13 6 (0, 1) and u0 > 0 throughout $2 exactly as in the proof of Theorem 4.1. Set v0 = u~; so v0 6 ~'+. Again, the functional v w-> K~z1(v) de__fpjz~ (vl/p) is strictly convex on f'+ U {0}. Recall that the subdifferential 0/Cz 1(v0) of K~zj at v0 exists in D' (S2) and is given by formula (4.5) with v0 in place of vl, ~ = )~l and f ( x , uo) - f ( x ) . Since u0 is a critical point for JZl, so is v0 for K~I. But this means that v0 is the global minimizer for K~Zl over V+ U {0} and the unique critical point of/C~ 1 as well. As a consequence,/Cz~ is bounded from below on ~'+ U {0}. On the other hand, from (4.3) we compute =/-)d~. 1

1) -- P

F

1/Pqgl

=

l / p fs? f gol dx

for r a R+, which shows K~z1(rqgf) --+ -cx~ as r --+ +c~, a contradiction to the boundedness of/Cz~ from below. The theorem is proved. 71 REMARK 4.3. In Theorem 4.2, the hypothesis 0 ~< f ~ 0 in ~2 is not necessary for the nonexistence in problem (4.6); it can be "slightly" perturbed, see [57], Corollaries 2.4 and 2.9, for the special case A (x, ~) = I~ Ip, (x, ~) E I2 x R N . We will treat this generalization later in Section 8.6 (Theorem 8.14).

Nonlinear spectral problems

407

4.3. Anti-maximum principle f o r )~ > ~1 We combine the Hopf maximum principle (Lemma 2.5) with the nonexistence result for )~ = ,kl (Theorem 4.2) to derive the so-called anti-maximum principle for the supercritical spectral problem - div(a(x, Vu)) = ~,B(x)lulP-2u + f ( x ) u=0

in ~2,

on0S-2,

(4.7)

with )~ - ~1 > 0 small enough, which is due to Fleckinger et al. [27], Throrbme 2, p. 732, or [28], Theorem 2.4, p. 55 (see also [56], Theorem 7.2, p. 154), again, for the special case A ( x , ~) - I~l p, (x, ~) e 1-2 • ]I~ x . The anti-maximum principle was first obtained by C16ment and Peletier [9], Theorem 1, p. 222, for the linear Dirichlet problem (p -- 2) using spectral analysis for )~ near )~l. THEOREM 4.4. Let s C ]t~N be a bounded domain with C 1'~ boundary f o r some t~ E (0, 1), and let s satisfy the interior sphere condition at every point o f 01-2. Assume that A and B satisfy Hypotheses (A) and (B), respectively, and f ~ L ~ (S-2) satisfies 0 <<,f ~ 0 in S-2. Then there exists a constant 6 =_ 6 ( f ) > 0 depending on f such that every weak solution u to problem (4.2) satisfies the anti-maximum principle u
in l2

and

OU

Ov

>0

(4.8)

on O~2

whenever )~1 < )~ < )~l + 6.

Recall that any weak solution u E Wo 'p ($2) to problem (4.2) with any )~ 6 ~ satisfies u 6 C 1'~ ( ~ ) for some 13 6 (0, or), by Proposition 2.1. PROOF OF THEOREM 4.4. We proceed by contradiction. Suppose there is no such constant 6 = 6 ( f ) > 0. Then there exists a sequence {ak}~-I C ()~l, cx~) with otk --+ )~1 as k --+ ~ , such that for every k = 1, 2 . . . . . problem (4.7) with )~ = otk has a weak solution uk E W I'p (S2) which does not satisfy inequalities (4.8). This means - div(a(x, V u k ) ) -- ak B(x)lUklP-2uk + f ( x ) U~=0 on0S2.

in s

(4.9)

We claim

Ilu~ll~

~

~

as k - - + c~.

(4.10)

Suppose not; then there is a subsequence of {uk}~__l, denoted by {Uk}~X~__1 again, that is bounded in L~(I-2). The regularity result in Proposition 2.1 implies that {uk}~__l is bounded in C 1 ' ~ ( ~ ) for some 13 6 (0, 1). Moreover, by Arzel~-Ascoli's theorem, {uk}~=l is relatively compact in C l't~* ( ~ ) for any 13" c (0,13). Thus, we may extract a convergent

P. Tahi(

408

subsequence Unk ~ U* in C 1'~* ( ~ ) as k ~ cx~. Letting k --+ cx~ in the weak formulation of (4.9), we arrive at

La(x,

Vu*).V~dx--~,lLB(x)[tl*lP-2u*wdx-}-Lf(x)llddx

for all w ~ Wo 'p (S2). So u* 6 C l't~* ( ~ ) is a weak solution of problem (4.7) with X = )~l, a contradiction to Theorem 4.2. We have proved our claim (4.10). Now set v~ -- uk/lluk]l~ for k - 1, 2 . . . . . obviously Ilvkll~ - 1. Thus, problem (4.9) becomes - div(a(x, Vvk)) = otkB(x)[vklP-2vk +

f (x)

in S2,

ilu~ll~-~

(4.11) on 3S-2.

v~ - 0

Since {Vk}L1 is relatively compact in C 1'~* ( ~ ) , let us extract a convergent subsequence

Vn~ --+ v* in C l'fi* ( ~ ) as k --+ c~. Again, letting k --+ c~ in the weak formulation of (4.11), we arrive at

.(x, vv*). V w d x

=

B(x)l

*lP-2

*wdx

for all w e Wo 'p (f2). We conclude that v* e C l'fl* ( ~ ) i s an eigenfunction for the nonlinear eigenvalue problem (3.2), with I[v* I1~ = 1. Hence, v* = c~ol with some constant c 6 R, c ~ 0. We distinguish between the following two cases: Case c > 0. There is an integer k0 ~> 1 such that each Vnk, for k ~> k0, satisfies the Hopf maximum principle (Lemma 2.5), that is,

Vnk > 0

inS-2

and

0 Vnk 3v

<0

on3S2.

(4.12)

We rewrite (4.11) as follows: - div(a(x, V Vn~))

- X1B(x)lvnk Ip-2vnk

-Jr-(Olnk -- )~l ) B (x ) [Vnk IP-Z Vnk -Jr- f(x)p-I IlunkI1~

Vnk - - 0

(4.13) in S2,

on O~2;k ~> k0.

Since

fnk (X)

de.._f(Olnk -- X1)B(x)lvn~ Ip-2 v ~ +

f(x)

> 0

in S2

Ilu~k IIp-1 ~ we may apply our nonexistence result, Theorem 4.2, to problem (4.13) with fnk in place of f , thus arriving at a contradiction.

Nonlinear spectral problems

409

Case c < 0. Again, there is an integer k0 >~ 1 such that each -Unk, for k/> k0, satisfies inequalities (4.12). But this contradicts our assumption that Unk = Ilunk II~ Vnk does not satisfy inequalities (4.8). Theorem 4.4 is proved. D REMARK 4.5. The hypothesis 0 ~< f ~ 0 in I-2 in Theorem 4.4 is not necessary for the (weaker) anti-maximum principle u < 0 in I2. It can be proved under the (weaker) hypothesis fs? fq)l dx > 0 combined with the nonexistence for problem (4.6) (i.e., the conclusion of Theorem 4.2). For the special case A(x, ~) = I~1p, (x, ~) ~ S2 x IRN, this improvement is due to Arcoya and G~imez [5], Theorem 27, p. 1908. We will give a different proof thereof later in Section 8.5 (Theorem 8.13). We have seen in the proof of Theorem 4.4 that the Hopf maximum principle (Lemma 2.5), called also Hopf's lemma, applied to the eigenfunction q91 (cf. Remark 3.6) played a crucial role in obtaining the anti-maximum principle (4.8). This fact has been explored further in greater detail for domains S-2 with nonsmooth boundary (e.g., with corners in R 2) and for p -- 2 in the works of Birindelli [6] and Sweers [54]. They studied the questions of the validity of Hopf's lemma and the anti-maximum principle separately. REMARK 4.6. The anti-maximum principle has played an important role in the recent studies on the Fu6~ spectrum of the p-Laplacian A p with various boundary conditions; the reader is referred to [ 11,27,28] and numerous references therein for analytical results, and to [8] for numerical results. REMARK 4.7. Last but not least, let us mention that many of the results presented in this section have been generalized to systems of equations involving the p-Laplacian A p. Most of them remain valid for strictly cooperative systems; see [10,28,30-33,56].

5. Linearization about the first eigenfunction We would like to answer the question of existence and uniqueness or multiplicity of weak solutions u E W~ 'p (S-2) to problem (4.2) also in the "resonant" case )~ -- ~,1. Recall that for p = 2 the problem in (4.2) is semilinear and has been extensively studied. In particular, if f (x, u) = f (x) is independent from u 6 • where f 6 L2 (~(2), then one can apply the standard Fredholm alternative for a selfadjoint linear operator on the Hilbert space L 2 (S2) in order to conclude that either (i) (f, ~01) = 0 in which case the set of all weak solutions to problem (4.2) is a straight line {u0 + rq91 "r ~R} in w~'P(I2) with (u0, q)l)--0, or else (ii) (f, q)l) 7~ 0 in which case problem (4.2) has no solution. Arguing by intuition from bifurcation theory, one should expect that the straight line of solutions from case (i) becomes "deformed" for p 7~ 2. To describe this phenomenon, another parameter (besides r c IR) has to be introduced into problem (4.2). In [22,24,58],

410

P. Tal~a

the orthogonality condition (f, gol) = 0 has been replaced by f__fll.qgi +fT,

-2 whereflldefllgolllLz(s?)(f, gol)and (f T , g ) l ) = 0 ,

(5.~)

with ~" = fll ~/t~ being the parameter and f T ~ LOO(S-2) fixed, f T ~ 0 in ,(2. Let u

Wo 'p (S-2), u = rq)l + u T with r ~ ]R, be a solution of (4.2). The following a priori relation between r and ~" was established in [58], Proposition 6.1, p. 331, for p 7~ 2" -2 lim (IrIP-2r~ ") = (p - 2)llgol IIL2(s2). Irl~oc

Qo(uo, uo) ~0,

(5.2)

where Qo(u0, u0) is a positive number depending on f T but not on ~" 6 ]R. This number corresponds to the quadratic form associated with the linearization of problem (4.2) about the first eigenfunction q)l, with ~. -- ~.l, described in Remark 3.6. In order to present this linearization and its important consequences in a tractable manner, from now on we restrict ourselves to the special case A(x, ~) = [~[P and B(x) = 1 for (x, ~) ~ I2 • tRN treated in [21,23,24,57-59] and many other articles. In particular, ~,j and q)l satisfy --ApgOl -- ~.ilgol]P-2gOl

in I2;

q91 = 0

on 0.(2.

(5.3)

The eigenvalue )~l is given by the variational formula (1.8). The eigenfunction go1 associated with )~l is normalized by ~01 > 0 in $2 and t[q)l IILP(S?) = 1. We would like to stress that practically all our results presented below apply to the general case as well, with the obvious necessary adjustments, provided A and B satisfy Hypotheses (A) and (B), respectively. We leave details to the reader.

5.1,

Linearization and quadratization

in order to determine the asymptotic behavior of "large" solutions u to problem (1.6), u = rq91 + u T with r ~ R and u T ~ w~'P(s2) as [rl --+ cx~, we will estimate the functional u v ~ ,.7Xl(r~Ol + u T) by suitable quadratic forms. Recall that ,.7xl has been defined in (1.4). To this end, we need to compute the first two Fr6chet derivatives of the functional ,.7xl. Our computations are rigorous for p > 2; we leave a few formal corrections for 1 < p < 2 to the reader. We define the functional .Y'(u) def = --1 fs? IVu] p dx, P The first Fr6chet derivative

u ~ w~'P(s2).

f'(u) of f" at u ~ Wo'P(s2) is given by H(u) = - A p u in

W - ~'P' (,(2), where ~1 + ~1 = 1. This follows from

( 7 (u), cA)-- is2 IVulP-ZVu " V~ dx,

q~~ Wo 'p ($2).

(5.4)

Nonlinearspectralproblems

411

The second Fr6chet derivative ~-fl (u) is a bit more complicated; if 1 < p < 2, it might have to be considered only as a G~teaux derivative which is not even densely defined:

(f"(u)0, e) -- ~

IVulP-2{(Vq~ 9 V 0 ) + ( p -

2 ) l V u l - 2 ( V u 9V4~)(Vu. V~p)} dx

'VulP-2(I-+-(p-2) ~,lVU| ~ 'Vu'

, v~ |

)

v~

RN• dx,

(5.5)

~,7~ e Wo'P(s2).

RNxN,

Of course, I is the identity matrix in the tensor product a | b stands for the (N x N)-matrix T whenever a - ( a i ) L 1 and b - ( b i ) L 1 are vectors

)•N•

(aibj)iN,j=l

RNxN.

from R N, and (., 9 denotes the Euclidean inner product in For a E R N (a = Vu in our case), a 7~ 0 6 R N, we introduce the abbreviation A(a)

def lalp_2(I+(p_ 2 ) a | a ) la12

.

(5.6)

If p > 2, we set also A(0) de__f0 E I[{NxN . For a 7~ 0, A(a) is a positive definite symmetric matrix. Its positive definite square root is equal to v/A(a)

de---flal(P-2)/2(I+ (-l + v/p -1) a | a) lal 2

9

The spectrum of the matrix [al2-PA(a) consists of the eigenvalues 1 and p - 1; moreover, we have A(a)v-

lalp-2v

for u

A(a)a-

(p - 1)lalP-ea.

E 1~N, with v" a - 0,

For all a, v E R N \ {0} we thus obtain

(A(a)v, V)RN min{1, p - 1} ~<

lalp_2lvl2

~ max{l, p - 1}.

(5.7)

Notice that for the general version of the energy functional (1.1) the matrix A(a) takes the form

A(xa)de--flalP-2(Oai) N , 77-- I(x,O~) g j -- - l a [ i,j=l P where x E s and ~ - - l a l - l a for a E •N \ {0}.

p-2

( 02A O~i O~j

(X, ~)) N

i,j=l

412

P Tak6(

From this point on, until the end of this paragraph, we restrict ourselves to p > 2. The case 1 < p < 2 is somewhat different and will be taken care of in detail in the second half of the next subsection (Section 5.2). We rewrite the p-homogeneous part of the energy functional (1.4) with ~. -- )~l using the integral forms of the first- and second-order Taylor formulas. Let 4) 6 Wo 'p (S2) be arbitrary. We combine (5.3) and (5.4) to obtain 1 fs21V(~Pl + P

=

~b)[P dx - )~lf~ 2 ](/91-+-~blP dx P

lifo /0'J

IV(~I -~- Sq~),P-2v(qgll + sO)" Vq~dx ds

-- )~1

(5.8)

1991 -+ S~[ p-2 (q91 --[-Sq~)q~ dx ds.

Similarly, using (5.3) and (5.5), we get (5.9) P

p

where Q~ is the symmetric bilinear form on the Cartesian product [Wo 'p (S'2)] 2 defined as follows, using the matrix abbreviation (5.6):

Q,(v, w) =

f ([f0'

]

A(V(qgl + s~b))(1 - s)ds Vv, Vw

- ~1 (P - 1)

) ]

dx

•N

[J0'

]qgl -~- $d/)[p-2 (1 - s) ds v w dx

(5.10)

for v, w E W01'p (S'2). In particular, one has

2. Qo(v, v)

-f (A(V~.)vv.vv)~.~-X.(p-1)f~f-2v2dx V~Ol

-- fs2 IVqgl]P-2 { ]Vvl2 -+-(p - 2) - ~l(P -- 1)

qgP-Zv 2 dx,

.Vv

2}dx

v 9 w~'P(s2).

(5.11)

Furthermore, our definition (1.8) of X1 and (5.9) guarantee Qt4 (q~, q~) ~> 0 for all t 6/t~ \ {0}. Letting t --+ 0, we arrive at Q0(4~, ~b) ~> 0

for all ~b E W~ 'p (S2).

(5.12)

Nonlinear spectral problems

413

Next we wish to show that the symmetric bilinear form Q0 is closable in L2(f2) and to characterize the domain D~0~ of its closure; see, e.g., Kato [42], Chapter VI, w1.3, p. 313.

5.2. The weighted Sobolev space 79~1 In the sequel, we always assume the following hypothesis on the domain $2" HYPOTHESIS (H1). If N ~> 2 then ~ is a bounded domain in ]~N whose boundary Ol2 is a compact manifold of class C 1,~ for some ot 9 (0, 1), and S-2 satisfies also the interior sphere condition at every point of 012. If N -- 1 then S2 is a bounded open interval in R l . It is clear that for N ~> 2, Hypothesis (HI) is satisfied if s C ]1~N is a bounded domain with C 2 boundary. The Hopf maximum principle (Lemma 2.5) guarantees (see Remark 3.6) ~01>0

inI2

and

0991 <0 Ov

onOI2.

(5.13)

We set

U de__.{x f 9 f)" Vq31(x) :)~ 0}, hence ~2 \ U -- {x 9 ~2" Vq91(x) -- 0}, and observe that ~2 \ U is a compact subset of ~ , by (5.13). We need to separate the cases 1 < p < 2 and p > 2. We start with the degenerate case 2 < p < cxz. Let us introduce a new norm on W1' p (I2) by

[II)[ID~1 de__f and denote by

~Z[)991the

[V~ol [P-ZIVvl2 dx

for v 9 W~ 'p (~2),

(5.14)

completion of Wo 'p ($2) with respect to this norm. That the

seminorm (5.14) is in fact a norm on wI'P(Y2) follows from inequality (5.18) below. The Hilbert space 79~Ol coincides with the domain of the closure of the quadratic form Q0" w~'P(s2) --+ IK given by Q0(4~) -- Q0(4~, 4~) for 4~ 9 wl'p(~2), cf. formula (5.11). The singular case 1 < p < 2 is more complicated. The Hilbert space D~Ol, endowed with the norm (5.14) for p > 2, needs to be redefined for 1 < p < 2 as follows. We define v 9 79~ if and only if v 9 Wo'2(~2), Vv(x) - 0 for almost every x 9 I2 \ U = {x 9 Y2" V~01(x) - 0}, and

Ilvllz~ de__f

179911P-21Vvl 2 dx

Consequently, D~01 endowed with the norm

< ~.

(5.15)

II 9 libel is continuously embedded into

Wo '2(S2). We conjecture that D~Ol is dense in L2(I2). This conjecture would immediately follow from IS2 \ UIN - - 0 . The latter holds true if S-2 is convex; then also S-2 \ U

P. Tak6(

414

is a convex set in ]t~N with empty interior, and hence of zero Lebesgue measure, see [28], Lemma 2.6, p. 55. If the conjecture is false, we need to consider also the orthogonal complement

~D-L'L2 = {v E L2(a"-d)" (V ~b)--Oforall~b~D991} 991

,

9

Notice that v E D/'L2 /'L2 991 implies v - 0 almost everywhere in U. This means that 7999 1 is isometrically isomorphic to a closed linear subspace of L 2 ( ~ \ U). Consequently, if v ~ _99~79 • and v is continuous in an open set G D ~2 \ U, then v = 0 in I2. Indeed, this claim follows from the fact that S2 \ U has empty interior, by (5.3) combined with (5.13). In "1~• contrast, if v E L 2 ( ~ ) satisfies (v, 91) # 0 then v ~ --991 9Furthermore, we have Xse\u ~'L 2 • 2 D99~, the closure of D99~ in L2(12), which implies that D99~ is isometrically isomorphic to a proper subspace of L 2 (~2 \ U). This can be seen as follows. Fix any e > 0. Since U t = 12 \ U is a compact subset of S2, and the Lebesgue measure is regular, there is an open set G C ]~U such that U t C G, G C 12, and IG \ U~IN <~e. In particular, 6 def dist(U', S-2 \ G) > 0. Now define K0 -- {x 6 ~ : dist(x, 12 \ G) <~ 3/3 }, K 1 - {x E ~ : dist(x, U') ~< 6/3}; hence, dist(K0, K1)/> 3/3. By Tietze's extension theorem, there exists a continuous function v'~2 ~ [0, 1] such that v - - 0 on K0 and v = 1 on K1. This function can be mollified in a standard way (using a convolution of v with a smooth nonnegative function with compact support in a ball of radius < 3/3 centered at the origin) to obtain another C 1 function w" 12 ~ [0, 1] such that w - 0 in an open neighborhood of 12 \ G and w = 1 in an open neighborhood of U ~. Clearly, w 6 D991 and

is2 Iw -Xu'12dx <~fG \u' dx <~e. It follows that Xu' 6 ~L2 991 as desired. Several important properties of D991 established in [57] are presented below. The following result is obvious [57], Lemma 4.1. LEMMA 5.1. Let 1 < p < cx~, p # 2, and let Hypothesis (HI) be satisfied. Then we have Q0(91, ~pl)=OandO<~ Q0(v, v) < cx~forall v E7999I.

5.3. A compact embedding with a weight for p > 2 We assume 2 < p < c~ throughout this paragraph. Notice that (5.7) entails

2 ~ fI2(A(Vqgl)V v , VV)RNd x <~ (p - 1)[[VII2v~ 1 IIv IIv~

for v E D99~.

(5.16)

Nonlinearspectralproblems

415

For 0 < 8 < oe, we denote by

~f~8 de___f{X E if2" dist(x, 0 s

< 8}

(5.17)

the 8-neighborhood of O12. Its complement in 12 is denoted by S2~ = S-2 \ 12~. The following compact embedding result was first proved in [57], Lemma 4.2, p. 199. LEMMA 5.2. Let 2 < p < cx~ and assume that Hypothesis (HI) is satisfied. Then we have: (a) For every 8 > 0 small enough, II 9 IIv~ is an equivalent norm on Wlo '2 (12~). (b) The embedding D~o~ ~

L 2 (I-2) is compact.

PROOF. Part (a) follows immediately from (5.13). To prove (b), we start with the proof of continuity of D~ ~ L2 (Y2). We take advantage of the Dirichlet boundary value problem (5.3) to compute, for every v E C 1 (~),

=2f

IV~pllP-2(V~ol" Vv)Vqgll d x - ~ .VqgllPv2qgl2 dx.

Adding the last integral and estimating the second to the last by the Cauchy-Schwarz inequality, we arrive at

)~lfs2q)f-2v2dx-k-fs 2 IVqgllpv2q)12dx ~< 2

<~ 2

IVq911P-2IVol2 dx

)1/2(fs2

IVq)l lp-2lVvl2 dx + -~

IVqgl IPv2q)12 dx

)1/2

IVq)lIPv2qgl2 dx,

and therefore,

)~1 fs? qgf-Zv 2 dx + ~1 fs2 IVqglIPv2 qg(2 dx ~< 2 Ilvllv~2

1.

(5.18)

Since C~ (S2) is dense in D~0~,the last inequality holds also for every v E D~l. Using (5.13) we conclude that the embedding D~01 ~ L 2 (~2) is continuous.

P Tak6(

416

To prove the compactness of 79~0~ ~ L2 (at'2), we take advantage of the Dirichlet boundary value problem (5.3) again to compute, for every v 6 D~I,

,~l fo ~ol~

lv2dx

-

L IV~~Ip-2v~~

2fn

V(V2) dx

IV~011P-llVvl 9 Ivldx

~< 211v1179~1

(f )1/2 IVq)lIPv2dx

(5.19)

by the Cauchy-Schwarz inequality. Let {Vn}n~__l be any weakly convergent sequence in D~Ol; we may assume Vn ~ O. Hence, Vn

~

0

weakly

in L2($2)

IVqgll(p-2)/2Vvn ~ 0

(5.20)

and

weakly in [L2($2)] N

(5.21)

as n -+ oo, where Vvn E [W-1'2($'2)] N . We will show that, indeed, Vn --+ 0 strongly in Lz(s2). Given any 0 < 0 < oo small enough, let us decompose $-2 = Uo tO U0' where = {x E n" ]VgOl(x) I > r/} Uo def

U,!def - {x ~ n" [Vqgl(x)] ~< r/}

and

(5.22)

We deduce from (5.20) and (5.21) that the restrictions Vn Iu, of Vn to Uo form a weakly convergent sequence in WI'2(Uo). It follows that IlVnlIL2(U,7) --+ 0 as n --+ cx~, by Rellich's theorem. Next, in (5.19) we replace v by Vn. Owing to (5.21), there is a constant C > 0 independent from n such that Ilvn IIz)~t ~< C~.~/2, and consequently, (5.19) yields

fs ~0p-1Vn2 dx ~
IV~ollPv~dx

),,2

(5.23) !

We split the integral on the right-hand side using S-2 - U0 t_J U,. The two integrals are estimated by

fU

r/

IVy011p2 Vn dx <~ IIV~o~IIp IV~ll Vn

fu

On

1/

(5.24)

Vn2 dx'

Vn

(5.25)

Now choose any 0 < e < cx~. First, fix 7/o > 0 small enough so that 8

0~/2" sup IlVnlIL2(~) ~ ~>~1 C~

9

(5.26)

Nonlinear spectral problems

417

Second, fix r / > 0 and 6 > 0 sufficiently small such that 0 < 77 ~< 00 and 12~ C U0, where the set $2~ has been defined in (5.17). This choice is possible by the Hopf maximum principle (5.13) for qgl. Third, recalling IlVnIILz(U,) --+ 0 as n --+ ~ , fix an integer no ~> 1 large enough such that for all n ~> no.

(5.27)

The numbers r/, 6 and no being fixed, we first apply (5.26) and (5.27) to (5.24) and (5.25), respectively, and then combine the last two with the inequality (5.23), thus arriving at fs_2 q9p - 1 v n2 dx ~< e

for all n ~> no

(5.28)

In particular, setting ~2~ = S-2 \ 12~, we infer from (5.13) and (5.28) that ]lvn Ilt2(s2~) ~ 0 asn---+

oo.

Finally, we make use of U0 U 12~ = 12 to conclude that proof of the lemma is finished.

Ilvn IIL2
as n ~ e~. The D

REMARK 5.3. For N = 1, L e m m a 5.2 follows from [37], Lemma 1.3, p. 238. Also the idea of using the bilinear form Q0 was introduced there. The case N = 1 is much simpler / to handle because one can compute the asymptotic behavior of the derivative ~p] (x) near its zeros very precisely, see (5.34) below. Now we are able to construct the closure of the symmetric bilinear form Q0 given by (5.11); see [42], Chapter VI, w p. 313. We extend the domain of Q0 to 79~o~ x D~o~. This extension of Q0 is unique and closed in L2(I2), as a consequence of inequality (5.16) combined with Lemma 5.2(b). Notice that the embedding W~ 'p (S-2) ~ D ~ is continuous, as qg] ~ C j ( ~ ) . We denote by A ~ the Friedrichs representation of the quadratic form 2 Qo in L 2 (S2); see [42], Theorem VI.2.1, p. 322. This means that A~o~ is a positive semidefinite, selfadjoint linear operator on L2($2) with domain dom(A~o~ ), such that dom(A~o~ ) is dense in 79~o~ and

(r

v, w) = 2. Qo(v, w)

for all v, w ~ dom(A~oj ).

Notice that our definition of Qo yields Ar compact, the null space of A~ol denoted by

- 0 . Since the embedding 79~1 ~

L2($2) is

ker(A~o, ) - { v ~ dom(A~o, ): A~, v - 0 } is finite-dimensional, by the Riesz-Schauder theorem [42], Theorem 111.6.29, p. 187.

5.4. Simplicity of the first eigenvalue for the linearization Also throughout this paragraph we keep our assumption 2 < p < c~. In addition to (H1), we impose the following technical hypothesis on the domain ~ .

418

P Tak6g

HYPOTHESIS (H2). If N ~> 2 and O~2 is not connected, then there is no function v 6 Dqgl, Qo(v) - O , with the following four properties: (i) v - 91"Xs a.e. in f2, where S C S2 is Lebesgue measurable with 0 < ISIN < l~2[u; (ii) S is connected and S N 01-2 ~ 0; (iii) every connected component of the set U is entirely contained either in S or in S-2\ S; (iv) ( a S ) A F2 C S2 \ U. It has been conjectured in [57], w that Hypothesis (H2) always holds true provided (HI) is satisfied. The cases, when ~ is either an interval in IR1 or else O1-2 is connected if N ~ 2, will be covered within the proof of Proposition 5.4 ([57], Proposition 4.4, pp. 202-205): We will show that there is no function v ~ D~01, Q0(v) = 0, with properties (i)-(iv). Lemma 5.1 provides another variational formula for ~,l, namely,

~,1 inf{f~ (A(V~ol)Vu,VU)]~Ndx " 0~U EDqgl}; -

-

(p - 1)

fo q)~-2lul2 dx

(5.29)

cf. (1.8). This is a generalized Rayleigh quotient formula for the first (smallest) eigenvalue of the selfadjoint operator (p - 1)-l,A~01 + )Vl~0~-2 on L2(S2). The following result determines all minimizers: A minimizer u 6 D ~ for )vl in (5.29) is unique up to a constant multiple of q)l. This statement is equivalent to" PROPOSITION 5.4. Let 2 < p < c~ and assume that both Hypotheses (H1) and (H2) are satisfied. Then a function u ~ ~)q91 satisfies Qo(u, u) = 0 if and only if u = Icq) 1 f o r some constant x ~ R. This result is due to [57], Proposition 4.4, pp. 202-205; its proof given below is quite technical. We stress that this proposition is the only place where Hypothesis (H2) is needed explicitly. All other results in this article depend solely on the conclusion of the proposition, that is, dim(ker(A~l )) = 1, which in turn implies (H2).

PROOF OF PROPOSITION 5.49 Step 1. Recall that the embedding D~o~ ~ L2(a'2) is compact, by Lemma 5.2(b). Let u be any (nontrivial) minimizer for )Vl in (5.29). First, suppose that u changes sign in S2. Denote u + -- max{u, 0} and u - - m a x { - u , 0}. Then we have, using [39], Theorem 7.8, p. 153,

)Vl--

f~ q9~-2(U+)2 fs2 qg~-2u2

fs2 (A(V~~

fl2 qgP-2(U-)2 ff2 99~-2u2

VU-q-)I~Ndx

(p - 1) fs2 qgf-2(u+)2

dx

f~2 ( A ( V g l ) V u - , VU--)I~N dx (p - 1)

p-2 ( u - ) 2 dx

Nonlinear spectralproblems

/> ( ff2 q)~)-2(U+)2 G q9f-2U2

+

419

fa,-2qOf-2(U-) 2 ) f ~ ~ol~__2U2

)~1

=X1. Consequently, both u + and u - are (nontrivial) minimizers for ~ 1 . Step 2. Let l; denote the set of all connected components of the open set U -- {x E ~2: V~0~(x) # 0}. We show that if u E ker(A~o~) then u is a constant multiple of ~o~ in each set V ~ F. Since q91 satisfies (5.3), it is of class C ~162 in U, by classical regularity theory [39],

def

Theorem 8.10, p. 186. Now, for each y E IR fixed, consider the function v z = u - y~o~ in ~ . Then both v + and v~- belong to ker(,A~ol) and thus satisfy the equation

--V. (A(V~Ol)Vvf) = ),.1(P -- 1)~0f-2V~ >/0 in U.

(5.30)

Again, by classical regularity theory [39], Theorem 8.10, p. 186, we have v•+ E C ~176 So we may apply the strong maximum principle [39], Theorem 3.5, p. 35, to (5.30) in every set V 6 V to conclude that either v + =- 0 in V, or else v + > 0 throughout V, and similarly for v~-. This means that sgn(u - y~01) -- const in V. Moving y from - o o to + o o , we get u = tcv~ol in V for some constant x v ~ IR, as claimed. Step 3. Let u E ker(A~ol ). Next we show that the fraction u/~01 "~2 --+ IR takes only finitely many values after u has been suitably adjusted on a set of zero Lebesgue measure. As above, for each y 6 IR fixed, consider the function ~• = (u/~pl) - y in ~2. We move y from - o o to + o o and use the fact that v~ - v• ~+ tpl 6 ker(A~0~ ) to conclude that fi0 =///991 must coincide with a finitely-valued function almost everywhere in ~2, because ker(A~0t ) is finite-dimensional and contains 91. Step 4. By contradiction, suppose that ker(A~oz) has dimension ~> 2. From ~01 E ker(,A~0~) and u E ker(A~01) ::> u + 6 ker(,A~0~) we deduce that there exists a function v E ker(,A~0~) with the following four properties: (i) v - ~pl 9 gs a.e. in ~ , where S C ~2 is a Lebesgue measurable set such that both S and ~ \ S have positive measure; (ii) the closure S is connected and S M O~2 # 0; (iii) for every V 6 1; we have either V C S or V C ~ \ S; (iv) x E ( a S ) N ~ ::> V~ol ( x ) - O. Indeed, such a function v can be easily constructed, starting from an arbitrary function u E ker(A~o~), u ~ Y~Pl for any y E IR" m

u(x)Aol (x) - ~

xi . Xs, (x),

xE~,

i=1 where K1 < K2 < " ' " < Km, m >~ 2, and every Si C ~ is Lebesgue measurable of positive measure, Si N Sj - 0 for i # j . Let us fix any y such that Xl < y < x2. For v• = u - ytpl we have v~- E ker(A~01 ) and the fraction v~- (x) v• (x) -

~ol (x)

- ( z - k:l) xs, (x),

x ~

420

P

Takd(

takes precisely two possible values a.e. on s namely, 0 and g - K 1 . Clearly, (/91 9 X S I E ker(A~01) and so (/91 9 X ~ \ S I - - q)l(1 - Xs~) ~ ker(A~Ol) as well; also, ISllg > 0 and lI2 \ S1 IN > 0. Replacing $1 by I-2 \ $1 if necessary, we may assume S1 N 0 I2 # 0. In particular, S1 possesses a connected component K such that K N 0 s # 0. The set S1 being compact, m m def we have K = S where S def S 1 0 K. It is obvious that the function v - q)l 9Xs is in ker(r ) and satisfies (i) and (ii); property (iii) follows from Step 2, whereas (iv) can be deduced from (iii) and inequalities (5.13). Step 5. Next, suppose also 0I-2 C S-. With a help from (5.13), this is equivalent to s \ S C X2. Choose any number k such that 0 < k < min x enSs~s91 (x), and define the functions ~ol~)d----ef min{qgl , k} Recalling q)l, l)

E

and

v(k) def = max{ v, q)l~) }

in S2.

ker(A~Ol), we have q9Ik) , v(~) 6 D~t together with v (k) = v . Xs + k. g n \ s

in $2. The Hilbert space D~o~ being the completion of wd'P(J2) in the norm II 9 IIzL~, there is a sequence {wn}~cc__l C w I ' P ( ~ ) such that IIw~ - vllz~, ~

0 and therefore also

Ilwn~k) - v(k) IIzLi -+ 0 as n -+ cx~, where

W(n~)defmax{wn, = q)Ik) }

in~2,n--l,2 .....

Here we have used the continuity of the mapping u w-~ u + "79~0t --+ Dq) 1 which is a version of Stampacchia's theorem; see [64], Theorem 1.56, p. 79. Set G = {x 6 I-2" qgl (x) > k} and observe that G D I2 \ S and I2 \ G C int(S). In view of Wn(~) = max{wn, k} in G, we have the inequality

~(

A ( V g o l ) V W n (k),

VW(nk))RN dx

_

-<

\a

(A ( V q g l ) V w ( k ) '

fG(A(V~ol)Vwn,

VW(nk))IRNdX --1-

VWn)RNdX.

Letting n --+ oe, we arrive at

fs2{A(Vq)l)Vv(~:), VV(~))~tN dx (A(Vq),)Vv (~0, Vv(")R u dx +

.< \G

f (A(V o,)vv,vv)N (5.31)

421

Nonlinear spectral problems Furthermore, in view of v (k) = v 9Xs + k . X~\s in ;2 and v - 0 < k in ;2 \ S, we have

fa,.2qgf-2(v(k))2dx--(fs-'l-fi2\S)qgf-2(v(k))

2dx

- f s qgf - 2 v 2 dx --{-k 2 f a,.2\ s qgl~- 2 dx

> fs-2q)f-21)2 dx.

(5.32)

We combine inequalities (5.31) and (5.32) with (5.29) to conclude that

)~1 ~< fs2 (A(Vq)l)Vv(k), vv(k))RN dx < fFe (A(Vq)I)VV, VV)It~N dx (p - 1) fs2 ~~

2 dx

(p - 1) fn ~~ -2v2 dx

=)~1,

m

which is absurd. Hence, we cannot have 012 C S which implies that 0S2 must not be connected. Step 6. With regard to our Hypothesis (H2), Step 5 leaves only the case N -= 1 still possible. This means that we may take I2 -- ( - a , a) to be an interval with 0 < a < oo, and (0, a) C S C [0, a) as well. However, the discontinuity of the function v at 0 is contradicted by the following embedding (and trace) result, L e m m a 5.5. This completes the proof of our proposition, that is, u -- x (/91 in X2 (x -- const), as desired. D LEMMA 5.5. Let 2 < p < oo and F2 -- ( - a , a) with some 0 < a < oo. Then there exists a constant C > 0 such that the inequalities

lu(y)-u(x)12

yl/(p-1) _ xl/(p-1) ~ (P -

c

1)

f0 a

fY

]u'(t)12t (p-z)/(p-1) dt

iu'(t)[2[~ ',(t)[ ~-2d/,

0 ~< x < y ~< a,

(5.33)

hold f o r every function u ~ 79~ ; in particular, the limit u ( 0 + ) = limx--,0+ u ( x ) exists. An analogous result is valid f o r the interval ( - a , 0). It follows that every function u E 79~9~ is H61der-continuous in [ - a , a]. REMARK 5.6. Unfortunately, no comparable result about the trace of a function u 6 ~)q)l on the set I2 \ U = {x e 12" Vq) 1 (x) = 0} is available for N ~> 2 as yet. This is the main reason why one needs to assume Hypothesis (H2) in Proposition 5.4. PROOF OF LEMMA 5.5. The Sobolev space Wo 'p ( - a , a) being dense in D ~ , it suffices to verify (5.33) for u e Wo 'e ( - a , a). Employing Cauchy's inequality, we compute for

P Tahi(

422

allO ~< x < y ~ a:

lu(y)- u(x)l

fx

Yuz(t)dt

(fY lUt(t)12t (p-2)/(p-1) dt )l/2(fxY

t - ( p - 2 ) / ( p - 1 ) d t ) 1/2

= (p -- 1)l/2(y 1/(p-l)- xl/(p-1)) 1/2

-')a,

which yields the first inequality in (5.33). The second inequality is obtained from the fact I that x ~01 (x) < 0 for all 0 < Ixl ~< a, and the following asymptotic formula:

]~o'l (x)]P-ego'1 (x) - -cx(1 + O(Ixl'+b))

as Ixl ~ o,

(5.34)

with b = 1 / ( p - 1) and a constant c = c(p, a) > 0. This formula can be obtained directly by integrating (5.3); see, e.g., [48], Eqs. (2.6) and (2.7), [37], Proof of Lemma 1.3, p. 238, or [47], Eq. (33), for details. E]

5.5. Another compact embedding f o r 1 < p < 2 In this paragraph, we switch to the case 1 < p < 2 and further require only (H1). In fact, Hypothesis (H2) always holds true in this case. Owing to ~ol 6 C l'~ ( ~ ) , for some/3 with 0 2, making use of Beppo Levi's equivalent characterization of Wo '2 (f2) quoted above. Notice that, by (5.7) for 1 < p < 2, (5.16) becomes /,

2 <<"Jr2" ](A(V~o1)Vv, Vu)~ N dx ~ IIv II2 (p - 1)llvllv~ "D~o1 and so Lemma 5.1 applies with no change.

for v 6 D~0~,

(5.35)

Nonlinear spectral problems

423

Next we highlight a couple of places at which the technique we use for p < 2 differs from that for p > 2. The most substantial difference between the two techniques is that the role of the compact embedding D ~ ~-+ L2(S2) needs to be replaced by that of W1'2 (I2) ~

7-/~, where 7-/~0~ is the Hilbert space defined below, 7-/~ ~

L 2 (I-2). Let us

define another norm on Wo '2 (I2) by def

IIv117%~ =

-2l) 2

qg~

dx

for v ~ Wo'Z(s2),

(5.36)

and denote by 7-/~x the completion of W1,2 (a'2) with respect to this norm. Embeddings that involve 7-/~o~are established next. They are taken from [57], Lemma 8.2, p. 226. LEMMA 5.8. Let 1 < p < 2 and let hypothesis (HI) be satisfied. Then we have: (a) The embedding 7-/~Ol~ L2(1-2) is continuous. (b) The embedding wl'2(I'2) ~ 7-/~Olis compact. PROOF. Part (a) follows immediately from (5.13). To prove (b), first notice that there exist constants 0 < Cl <~ c2 < oo such that Cl <~qgl ( x ) / d ( x ) <, c2 for all x ~ ~2, where the function m

d(x) de..~.fdist(x, OI2) -- inf Ix - x0l, xoeO~

x E S'2,

denotes the distance from x to 012. By well-known results taken from [43], w or [63], w or simply by an inequality similar to (5.18), the Sobolev space Wo'2(s is continuously embedded in the weighted Lebesgue space L 2 ( ~ ; d(x) -2 dx) endowed with the norm IIv IIt2~S~;d
(fo

0 2 d (x)-2 dx


Notice that 7-/~ol = L2(S2; d ( x ) p - 2 d x ) . Consequently, using again the splitting S2 = I2~ U 12~ from the proof of Lemma 5.2, we conclude that the embedding W~ '2 (I2) ~ is compact.

7-/~ol []

5.6. A few geometric inequalities In Sections 5.2-5.5 we have shown the most relevant properties of the quadratic form Qo(v) = Qo(v, v) defined in (5.10) and those of its domain, the Hilbert space D~o~. In the sections to follow, we often need to compare the quadratic form Q~(v) = Q4~(v, v) defined in (5.11) with Qo(v) = Q0(v, v), at least for 4) E w~'P(I2) n c l ( ~ ) . A natural way to do this is to compare the kernels of these quadratic forms, so that we can use

P.Tak6(

424

the Hilbert space Dr not only for Q0 but also for Q~. To this end, we will use the following elementary, but important geometric inequalities due to Tak~i6 [57], Appendix A, pp. 23 3-235. Recall that R+ = [0, cx~). We begin with the following auxiliary inequalities [57], L e m m a A.1, p. 233: LEMMA 5.9. Let 1 < p < c~ and p # 2. Assume that 69 ~ L ~ (0, 1) satisfies 0 ~ 0 in (0, 1) and T -- fd 69(s) ds > 0. Then there exists a constant Cp(O) > 0 such that the following inequalities hold true f o r all a, b E RN : I f p > 2 then

(Cp(O)))p_2( max

0~
la + sbl )p-2 ~< fo 1 la + s b l P - 2 0 ( s ) d s ~< T . (\O~
(5.37)

and if 1 < p < 2 and lal + Ibl > 0 then

[

T . [ max Ia + s b l k 0~
p-2

~< fo 1 la + sblp-269(s) ds

max [a + sb[ O~
(5.38)

PROOF. Only the inequalities involving the constant Cp(69) are nontrivial. Set q -- p - 2; hence - 1 < q < c~, q # 0. We prove the following weaker inequality first, with some constant x > 0:

fo 1[a -+- sb[ q 0

(s) ds

t 1/q >/x [a[

for all a, b 6 I~u.

(5.39)

The case a = 0 is trivial; so we will always assume a 6 R N \ {0}. Owing to the rotational invariance of the Euclidean norm in ]1{N, we may restrict our attention to the plane ~2 (N = 2) with a = (al, 0) 6 R 2 and b = (bl, b2) ~ R 2. Moreover, the homogeneity of both sides in (5.39) allows us to assume a - el def - (1,0) 6 ]1~2. We need to distinguish between the cases q > 0 and - 1 < q < 0. Case q > 0. Consider the function F : R 2 --+ ~ + defined by

def f 1

F ( b ) -- ]0 [el + sb[ q Og(s) ds

for b ~ R 2.

This is a continuous function which satisfies

, qf,

F ( b ) ~> -~b

S q 69(S)ds

for 0 < ~r ~< 1 and Ibl/> 2/or.

(5.40)

Nonlinearspectralproblems

425

This follows from 1 whenever 0 < cr ~ s ~< 1 and :cr lbl ~ 1. 2

1 lel + sbl ~ : s l b l Z

Taking into account 0 ~< O ~ 0 in (0, 1), we find and fix a number a 6 (0, 1) such that f l s q O ( s ) d s > 0. Consequently, by (5.40), F possesses a global minimum which is at0 tained at some b0 E ]~2. It remains to show that x def -- F(b0) 1/q > 0. Indeed, F (bo) would force el + sb0 = 0 E ]1~2 for almost every s 6 (0, 1) such that O ( s ) > 0, which is impossible. This proves (5.39) for q > 0. Case - 1 < q < 0. We observe that inequality (5.39) is valid if and only if def C - xq

fo lel + sbl q O ( s ) ds

for all b E ]~2.

Since 69 6 L ~176 (0, 1), it is obvious that it suffices to show this estimate for 69 =- 1 in (0, 1) and for all b = (bl, 0) 6 ]R2, that is, 1

C - sup bl cR

f0

l1 + sbl

Iq ds

< e~.

Indeed, for b l ~> - 1 we have fo I I1 -t- sbl Iq ds <<.fo I (1

- s) q ds

-- 1 +1 q

For b l < - 1 we estimate s

yll

I1 + sbll q ds = Ibl1-1 =lbll- 1

I1 - t l q dt 1 l+q

[ 1 - q - ( l b l l - 1 ) l+q] <

2 l+q

Ibll q <

2 l+q

Hence, inequality (5.39) is valid also for - 1 < q < 0. Next, in (5.39) we replace the function 69 (s) by O(s) ae__f69 (1 - s) for 0 ~< s ~< 1 to get the following new inequality, with some other constant ~ > 0:

fo 1 la +

sbl q O ( s ) as )

1/q ~

~ lal

for all a, b 6 R N. m

m

Replacing the pair (a, b) by (a + b, - b ) , and the function O ( s ) by O ( s ) - 69(1 - s), we have also

fo I l a + s b l q O ( s ) d s ) 1/q ~>t?la+bl

f o r a l l a , b ~ I R N.

(5.41)

426

P Tak6("

Finally, we take advantage of the convexity of a norm to deduce the desired inequality

fo 1 l a + s b l q O ( s ) d s ) 1/q ~r from (5.39) and (5.41), where

r

1/q" max

0~
(O) demfmin{x, t~} >

For instance, one gets optimal constants

la+sbl

0.

C2((~90)--

1/2 and

D

C2(O1) -- (3~/~) -1

for

Oo(s) = 1 and O1 (s)-= 1 - s, respectively.

We are now able to estimate the quadratic form associated with the symmetric matrix A(a) defined in (5.6). The inequalities below follow from a combination of (5.7) with (5.37) for p > 2 and (5.38) for 1 < p < 2, respectively. We omit the index ]~U for the Euclidean inner product (., .) in ]1~ N . LEMMA 5.10. In the situation o f Lemma 5.9, we have f o r all a, b, v

(Cp(O))p_2 (

fo'

max la + sbl 0~
~ ~,~N.I f p

> 2 then

)p-2 Ivl 2

(A(a + sb)v, v)O (s) ds

~<(p-1)T.

max l a + s b l O~
)p-2

Ivl 2,

(5.42)

and if 1 < p < 2 and lal-+- Ibl > 0 then

(p-1)T.

max l a + s b l O<~s~
)p-2

Ivl 2

1

(A(a + sb)v, v)O (s) ds

~(Cp(O))p_2(

max la + sbl O~
)p-2 Ivl2.

(5.43)

Finally, to estimate the quadratic form Q4~(v)= Q4(v, v) defined in (5.11), take Ol(S) _= 1 - s (0~ 2 then, by (5.42), for any v 6D~Ol inequality (5.16) is replaced by (Cp(Ol))p-2lloll 2

D~o1

~<

fo([f0'

] )

A(V(qgl + s~b))(1 - s ) d s V v , V v dx <<,cx~.

(5.44)

427

Nonlinear spectralproblems

On the other hand, if 1 < p < 2 then, by (5.43), for any v E D~0~ inequality (5.35) is replaced by

fs?([f01A(g(gol + s~b))(1-

(Cp(O1)) p-2

s) ds]Vv, Vv)dx

Ilvll~l.

(5.45)

6. An improved Poincar~ inequality for p > 2 We are now equipped with most of the technical tools we need to establish the existence of a weak solution u E W0, p (Y2) to problem (1.7) in the "resonant" case )v = and under the condition that f 6 L2(/2) satisfies (f, gol) = 0 .

6.1.

)~1

for 2 < p < ec

Statement and proof of Poincard's inequality

Recall the decomposition (5.1) of a function u ~ L2 (a'-2) into the orthogonal sum u - u II 9gol + uT

where

u Hdef 11
-2

(u , go,)

and

(u q- gol) = 0.

(6.1)

We motivate our approach by the following well-known inequality that follows directly from the spectral decomposition of the Dirichlet Laplacian A in L2(/2):

fa"2IVu]2dx- ~'' ~ lu[2dx > ()~2- ~'1)f~2 luTI2dx

(6.2)

for every u E WI'2(~Q). Here, it suffices that s be a bounded domain in •N (N ~> 1), and )vl and )v2 stand for the first (smallest) and second eigenvalues of --A, respectively, so that 0 < )vl < )v2. As a consequence of this inequality, one obtains immediately the "existence" part of the Fredholm alternative for the positive Dirichlet Laplacian - A at the first eigenvalue )Vl. In the work of Fleckinger and Tak~i6 [34], Theorem 3.1, p. 957, the power p -- 2 was replaced by any power p ~> 2; the Poincar6 inequality (6.2) was thus extended to the "degenerate" case 2 < p < oc.

Assume that both Hypotheses (H1) and (H2) are satisfied. Then there exists a constant c =- c(p, Y2) > 0 such that for all u E Wo 'p (Y2), THEOREM 6.1.

41

s? lVulP dx

-

1.1 fs? lulp dx

>>,c(lu"lP-2f~ IVgo,IP-2IVuT ]2

428

P Takd(

If the constant c in (6.3) is replaced by zero, one obtains the classical Poincard inequality; see, e.g., [39], Ineq. (7.44), p. 164. We call our inequality (6.3) an improved Poincard inequality. Estimating both integrals on the fight-hand side in (6.3) from below, we obtain [34], Corollary 1.2, p. 953: COROLLARY 6.2. Let ~ be as in Theorem 6.1. Then there is another constant c ~ =_ c'(p, I2) > 0 such thatforall u ~ w;'P(Y2),

fs? lVul p dx - ~l ff2 lulp dx (6.4)

This inequality trivializes to (6.2) in the "regular" case p - 2, with the optimal (largest possible) constant )~2 -- )~1 on the right-hand side being replaced by another positive constant 2c' ~< )~2 -- )~1. REMARK 6.3. Except when u II = 0, we may replace u ~ Wo'P(s

by v -- u / u II in in-

equality (6.3) and thus restate it equivalently as follows, for all v T 6 Wo'P(K2) with (l)-l-, q)l ) = 0 : c p(ll TII 2 + II TII

(6.5)

This remark indicates that our proof of inequality (6.3) should distinguish between the cases when the ratio ][uTllw2,p(se)/lull i is bounded away from zero by a constant y > 0, say, 1 3

Ilu w II wd,p (s~)/> y, lulll and when it is sufficiently small, say,

IluT II Wo',P(S~) ~ 0 is small enough. The former case is treated in a standard way analogous to (1.8), whereas the latter case requires a more sophisticated approach based on the second-order Taylor formula (5.10) applied to the expression Qvv(v T) on the left-hand side in (6.5) where v - u / u II. For either of these cases we need a separate auxiliary result: We derive two formulas for Rayleigh quotients outside and inside an arbitrarily small cone around the axis spanned by q)l, respectively.

Nonlinear spectral problems Minimization outside a cone around ~1.

Given any number 0 < y < ec, we set

Cg dem-~f{U E wI'P(~Q) 9 [[uT[[w~3,p(~) ~ ! def

1,p

T

YluJ l},

> },,lUll[ }

and C•l is the closure of C•c the complement

Notice that C• is a closed cone in W;' p (s of C• in W;' p (s

429

We consider also the hyperplane

('l

C~xzde2{u E W; 'p(~('2)" u II - - 0 } - -

0
For 0 < y ~< oo we define fs2 IVul p d x , def A• = inf, f o lulp dx " u e C• \

}

{0} .

(6.6)

LEMMA 6.4 ([34], L e m m a 5.1, p. 963). L e t 1 < p < oc and 0 < y <~ oo. Then we have A• > ~,1. PROOF. Assume the contrary, that is, A• = )vl for some 0 < y < oo. Pick a minimizing sequence {Un}n~ in C~ such that

fsl

unlPdx-

1

and

fs? IgunlPdx ~ ~,1

as n--+ oc.

Since W I'p (X2) is a reflexive Banach space, the minimizing sequence contains a weakly convergent subsequence in Wo'P(s which we denote by {Un}neC_] again. Consequently, Un --+ u strongly in LP([-2), by Rellich's theorem, and Vun ~ V u weakly in [LP([2)] N as n --+ oo. We deduce that fo lulp dx - 1 and

1/p )v 1 <<.IlVullLp(s2) <. liminf]lVunllLp(s2) -- xI/Pl

rl--+(X)

As the standard norm on the space W~ 'p (s

is uniformly convex, by Clarkson's inequal-

ities, we must have Un --+ u strongly in W o ' P ( ~ ) , by the proof of Milman's theorem (see [66], Theorem V.2.2, p. 127). This means that -2 (U , qO1) = II ol II-2 L2(/2) (Un, ~O1) ---> u li - II~Ol[Ig2(~)

u,T = u, -- U~ ~01 --+ u T - u as n --+ r

_

ull

~01

strongly in W~ p ( ~ ) ,

! ! The set C• being closed in W I'p ( ~ ) , we thus have u 6 Cy.

430

P. Tak6(

On the other hand, from IlUllLp(n) = 1 and IlVullLp(n) = ~.ll/P, combined with the simplicity of the first eigenvalue ~.l, one deduces that u -- +~Ol, a contradiction to u ~ C•t. The lemma is proved. D

Minimization inside a cone around q91. For 4) 6 Wol' p (f2), 4) ~ 0 in f2, let us define clef =

liminf

fn([fg A(V(gol + s~b))(1 - s) ds]V~, V~)RN dx

"*"w~'P(~)---'~

fn[fd

(6.7)

I~01+ s4~lP-2(1 - s) ds]lq~[2 dx

(~),~l)--0

with the abbreviation (5.6). Using the quadratic form Qr defined in (5.10), we notice that

X - - ~, 1 ( P - - 1 ) - -

lim inf 114~llw~'P(n)~0 (~,q~)=o

Q~ (4))

fn[fo I~0~+ sq~lp-2(1 -

~> O. s) ds]14~[2 dx

LEMMA 6.5 ([34], Lemma 5.2, p. 964). Let 2 < p < o0. We have A > ~l (P - 1). Before giving the proof of this inequality, we introduce the following notations where t 6 ~ and 4) 6 W~ 'p (S-2)"

o(t Pl(t,

effo[J~

[gOl + stqblP-2 (1 -- s)dslq~2 dx,

~) de=f( If01 A(V(qgl + stq~))(1- s)ds]V4), V~b).

Hence, expression (6.7) takes the form A=

liminf

pl (t, 4~)

with any fixed t 6 R \ {0}.

(~,~o~)=o Furthermore, owing to inequalities (5.37) and (5.42), the expressions P0(t, 4)) and 791(t, 40, respectively, are equivalent to .M0(t, 4)) de_f~n (~~

+ [tlP-2[~[P-2)~2 dx

= fn qgf-2q~2dx

-[- ]tlp-2llcbllPLp(n)

Nonlinear spectral problems

431

and J~l (t, t~) de2 fl2 ([Vg011P-2 + Itlp-2lVqblp-2)lVqb]2 dx

- tl4~l]2 --

D~i

+ ]tlP-2lldpl[ p

W~ 'p (S2) '

that is, there are two constants c l, c2 > 0 independent from t and 4~ such that ClJ~/(t, ~) <~ 79i (t, ~b) ~< c2J~/" (t, t~),

(6.8)

i = O, 1.

PROOF OF LEMMA 6.5. On the contrary, assume that A 4 )~l (P - 1). Pick a minimizing sequence {~bn}n~ in W~ 'p (12) such that q~n ~ 0 in S-2, (q~n, qgl) -- 0, Ilq~nIIWo~,~(s2) ~ 0, and 79~ (1, ~bn) 790(1, q~n)

--~ A ~ X1 ( P -

1)

as n ---~ oo.

Next, set tn = T0(1,4~n)1/2 and Vn = C~n/tn for n -- 1, 2 . . . . . Hence, we have tn --+ O, 790(tn, V n ) - 1 and 791(tn, Vn)--+ A as n --+ cx~. Inequalities (6.8) guarantee that both sequences IlVnllZ~, and tn1-(2/p) ]]Vnllwl,p(s-2)are bounded, and so we may extract a subsequence denoted again by {Vn}n~=l such that Vn ~ V weakly in D~01 and tl-(Z/P)gn .._x Z weakly in W; 'p (12) as n --+ cx~. Using the embedding W; 'p ( S 2 ) ~

79~1, we get z-= 0

in 12. Furthermore, both embeddings 79~o~ ~ L2(S2) and W~ 'p (12) ~-+ L p ( ~ ) being compact by Lemma 5.2(b), and Rellich's theorem, respectively, we have also Vn --+ V strongly in LZ(~Q) and tl-(2/P)Vn ~ 0 strongly in L p (S'2). It follows that (V, ~01) = 0 and 79o(0, V ) = ~

qgP-2v2 dx = 1,

7~1 (0, V) -- ~1(A(Vqgi)VV, VV) ~ A ~ X1 ( P -

1)

Consequently, Proposition 5.4 forces V = xq91 in s where x E N is a constant, x :/: 0 by To(0, V) = 1. But this is a contradiction to (V, qgl) = 0. We conclude that A > )~l (P - 1) as claimed. IS] PROOF OF THEOREM 6.1. If u ~ W~ 'p (X2) satisfies (u, q)l) = 0, then (6.6) implies

s [VulP dx - ~l ]" lul p dx

( z, )f lVulPdx_(l-A)~1~ ) fS2]vuT]pdx

>>. 1 - A ~

(6.9)

P TakC(

432

where ~,1/Aoo < 1 by Lemma 6.4. Thus, we may assume (u, ~1) ~ 0 and so we need to prove only inequality (6.5). We will apply Lemmas 6.4 and 6.5 to the following two cases, respectively. Case [IvYllw),p(s2) >~y. Here, y > 0 is an arbitrary, but fixed number. In analogy with I J

inequality (6.9) above, we have

L Iv~~+ vvwI~

- ~'1

~ I~Ol+ v T I~ d~

>/ (1-Ay~'l)fs2]VgOl+VvY]Pdx>/c~,fs2]VvT[Pdx for all 1)y E

Wo'P(ff2) such

that (I)T,~01) = 0 and

(6.10)

IlvTllwl.p(s2) >>,y, where cy > 0 is u

a constant independent from v y. The last inequality follows from the boundedness of the orthogonal projections u w-> u II .~Ol and u ~ u y in W~)'p (Y2). Recalling the embedding

Wo'P(F2) r

D~Ol, we deduce from (6.10) that inequality (6.5) is valid provided

IlvVllw~,~ y. Case IIvTIIwl,~ 0 is sufficiently small. According to (6.7) and u

Lemma 6.5 we have Qvz(v T) -- 7~1 (1, v T) - )~1 (P - 1)~0(1, /> ( 1 - X I ( P ~7- 91 1) )( 1 A

' vT)

/>c..A/'I(1, v T) for all VT E

wI'P(~Q) such

v T)

that (I)T, qgl) - - 0 and

(6.11)

IIvTIIw~,r(s~) ~< y, where y > 0 is suf-

ficiently small and ~ > 0 is a constant independent from v T. Recall that the expressions 7~i (1, v T) and .gr/(1, v T) (i = 0, 1) have been defined after Lemma 6.5. From (6.11) we deduce that inequality (6.5) is valid also when IlvT Ilw~,p(s2) <~y. Finally, in order to prove inequality (6.4), we make use of the embeddings D~l r L2(/-2) and Wo'P(Y2) r our proof of Theorem 6.1.

LP(a'2) to estimate the right-hand side in (6.4). This finishes ISl

REMARK 6.6. Recall that p > 2 and wI'P(F.2) r "D~Olr L2(~). Let f(x, u) = f(x) be independent from u ~ IR where f ~ LZ(a"2) satisfies (f, ~01) -- 0. Although the functional JZl defined in (1.4) is no longer coercive on W~)'p ( ~ ) , it is still not only bounded from below, but also "very close" to being coercive on the weighted Sobolev space D~01, as a direct consequence of improved PoincarC's inequality (6.3). This property of Jz~ will be used in the next paragraph to derive an existence theorem for problem (1.7) when ;v -- )v1.

Nonlinear spectralproblems

433

6.2. Fredholm alternative at )~1 In analogy with the case p = 2, inequality (6.3) guarantees the solvability of the Dirichlet boundary value problem

--/kpU -- ~I[u]P-2u %- f (x)

in s

u= 0

on 0s

(6.12)

in the following special case: THEOREM 6.7. If f E D~I : satisfies ( f , q31) = O, then problem (6.12) possesses a weak

solution u E W~' p (s This theorem is due to Fleckinger and Tak~i~ [34], Theorem 3.3, p. 958. Here we have denoted by D~0~ the dual space of D~o~ with the duality induced by the inner product (.,-) in L 2 (s We have taken advantage of the fact that the Hilbert space D~I is continuously and densely embedded in L2(s see Lemma 5.2(b). Hence, also the embedding L2(s ~ 7D,p : 1 is continuous. Notice that a sufficient condition for f E D~o~ is f E W-1,2 (s and f [~ E L2 (G) in some open set G D s \ U. The orthogonality condition (f, ~ol) -- 0 is sufficient, but not necessary to obtain existence for problem (6.12) provided p r 2 (1 < p < ~ ) , according to recent results obtained in [22], Theorem 1.3, for N = 1, in [24], Theorem 1.1, for any N ~> 1 and 1 < p < 2, and in [58], Theorems 3.1 and 3.5, for any N ~> 1; see also [21], Theorems 1.1-1.3. PROOF OF THEOREM 6.7. Our proof of this theorem combines the improved Poincar6 inequality (6.3) with a generalized Rayleigh quotient formula. To this end, we may assume that f 6 D t satisfies f ~ 0 in s and (f, ~01) = 0. Define the number M f (0 ~ MU ~ oo) by

a~r My --

I(f, v)IP

sup . ,cw:'"<~) fs~ [Vvlp dx --/kl fs2 Iv[P dx vr

(6 13)

:KER}

Clearly, M f > 0. Moreover, (6.3)entails

)J:li" w-~,p ' (s2)

If" w~'p (:2)

<~c : ( f lVvlP dx - X, f lvlp dx) for all v ~ w:'P(s

'

where C f = c -1 Ilfl[ p

w-l,p

' (s2)

is a constant. This shows that

P. Takdd

434

Mf ~ Cf < oo. In a similar way we arrive at

I,-,' I'-=ll= ~< Iv" I~-=ll.fll.~s, II~'~ I1='Dgl Cf where

C f, - c -

IVvl p dx - ~1

Ivl p dx

)

for all

v E w;'P(I2),

(6.14)

1Ilfll~o 1 is a constant, and II 9 IIV~l, stands for the dual norm on D'q919

From (6.13) and inequality (6.14) we can draw the following conclusion: If v ~ is such that v m ~ 0 in ~ and I(f, v)l p

WI'p (~)

1

f n lVvlP dx - ~.l f n lvlP dx

>~ -~M f ,

then (f, v) -r 0 and C}

I~"l ~-2 ~< 2 ~ l < f , where

p-2

~>1

-<

tt

(cf) ~-2

-2 II~T IIpw4,~)'

CUf ! = [2(C}/Mf)]l/(p-2)llfllw_l.p, (s2) is a constant, i.e.,

Iv'l ~
(6.15)

Next, take any maximizing sequence {Vn}n~176in quotient (6.13), that is, vnT ~ 0 in S2 and

I(f, Vn)l p fs2 IVonl p dx

- ~,l fr2 IVnlp dx

---->Mf

W; 'p ( ~ ) for the generalized Rayleigh

as n -+ oo.

(6.16)

Since both, the numerator and the denominator are p-homogeneous, we may assume

IlVnllw~,p(s?) - 1 for all n ~> 1. The Sobolev space w;'P(I2) being reflexive, we may pass to a convergent subsequence Vn ~ w weakly in W~ 'p (I2); hence, also Vn ---> w strongly in L p ( ~ ) , by Rellich's theorem, and (f, Vn) --> (f, w) as n ---> oo. We insert these limits into (6.16) to obtain

fs_e lVwlPdx-~.l fse lwl pdx ~ l --)~l fs_e lwlPdx = Mf~l(f, w)lP

(6.17)

In particular, we have w ~ 0 in s therefore also w T ~ 0 by (6.15), and consequently I(f, w)l 7~ 0 by (6.17). We combine (6.13) with (6.17) to get fs? IVwlp dx = 1. Hence, the supremum Mf in (6.13) is attained at w in place of v.

Nonlinear spectral problems

435

Finally, we can apply the calculus of variations to the inequality

fnl

Vv[P dx -)~1 fs? [v[pdx - M f l[(f, v)[ p >10

for v E W; 'p (X?)

to derive --Apto

w=0

-- ) ~ 1 1 t o l P - 2 w ~-~

Mill(f, w)lP-2(f, w ) f ( x )

in X?,

onO~. def

1 / ( p - 1)

It follows that u = M J Theorem 6.7 is proved.

(f, w)

- 1

9 w is a weak solution of problem (6.12).

6.3. Application to the embedding W o' p ~

D LP

The "geometry" of the Sobolev embedding W;' p (X?) ~ LP (X2) for p -- 2 is easily described by the eigenvalues {)~k}~_-l, with 0 < )~l < )~2 ~ ~.3 ~ ' " ", and the associated eigenfunctions {q)k}~=l, with (~Pk,r -= 1 and (~0k, q0e) - - 0 if k 7~ s for the positive Dirichlet Laplace operator - A in L 2 ( ~ ) . Simply, the unit sphere from W0'2(s after having been embedded into L2(~Q), becomes an (infinite-dimensional) ellipsoid with the axes of length ;~k 1/2 in direction q)k (k = 1, 2 . . . . ). Such a clear geometric picture is unknown for p r 2. Only the first two eigenvalues of the nonlinear operator - A p are known to have a variational characterization: )~1 by formula (1.8) and ~.2 by a minimax formula [3], Remarques 2.2, pp. 15-16, and 0 < ~.1 < )~2- A divergent sequence of eigenvalues 0 < )~l < ~.2 ~ )~3 ~ " ' " of - - / k p , characterized by a minimax formula, has been obtained in [3], Remarques 2.2, pp. 15-16, as well, but it is unknown if these are all eigenvalues of - A p . It is shown in [4], Proposition 2, p. 5, that there is no eigenvalue in the open interval ()~1,)~2). A weaker result, namely, that there is no eigenvalue in some open interval (,kl,)~1 + 6), 6 > 0, was obtained earlier by Anane [2], Thdorbme 2, p. 727. Using the results and methods from this section, we would like to address the problem of geometry of the Sobolev embedding w;'P(g2)~-~ LP(ff2) for p > 2: a kind of "stability" and nonsimplicity of the first eigenvalue )~1. More precisely, let us "squeeze" (deform) the unit sphere in LP(~C2) along a fixed vector f 6 W-I'P'(s orthogonal to ~01, that is, consider a new norm on W; 'p (~2) defined by def

IlullL~(s~)~I = (llull pLP(~?) -}- [(U, f)]p)l/p

1,p

for u e W 0

(X?),

(6.18)

where f 6 W - 1,p' (f2) is a given distribution from the dual space W - 1, p' ( ~ ) of W;' p (X2), with (f, ~01) - 0 . Of course, if f E LP'(aC2) then ][. [[Lp(S?);U is an equivalent norm on LP(a"2). Clearly [[~ol[[Lp(s?);/= [[r [[Lp(~?) = 1. Next, define a Rayleigh quotient analogous to (1.8), ~ f - - inf

[Vul p dx" u ~ w;'P(s

with

I[ullLp(~).f- 1 .

(6.19)

P Takdg

436

Observe that # f

~

~,1. On the other hand, if f ~ 0 in ~ then improved Poincard's inequal-

ity (6.3) guarantees for all u 6 Wl ' p (X2),

fx2 lVulP dx - ~l fs2 lul p dx

c f Igu- lPdx >>.cllfll -p W-I'P

I (if2)

9 ](u f)l p

Recall that c =_ c(p, ~ ) > 0 is a constant. Thus, we have proved the following result: LEMMA 6.8. If f ~ W - I ' P ' ( f f 2 ) s a t i s f i e s

IIfllw-~,/
(r

1/p and (f, ~Pl)--O, then

# f =)~l. Clearly, the infimum in (6.19) is attained at u = --l-q91 . In addition, our proof of Theorem 6.7 guarantees that this infimum is attained also at a point different from -+-~01,provided ! f is restricted to 7?~o1 \ {0}. PROPOSITION 6.9. Assume that f E D~o ' ~ satisfies 0 < IIflIw-~,/(s~) ~ (c/)~1) lip and (f, ~ol) =0. Then # f = )~l and the infimum in (6.19) is attained at • and anotherpoint uo E w~'P(f2), uo ~ -t-(pl in s PROOF. According to the proof of Theorem 6.7, the supremum M f (0 < M f < ~ ) defined in (6.13) is attained at some w ~ W~ 'p (f2) in place of v, with w ~ {x~ol" x 6 R}. Of -1

course, in (6.13) we may replace w by u0 = ]]W]lLp(U2);fW; hence [luo[ILp(ff2).f : 1. W e combine formulas (6.13) and (6.19) with Lemma 6.8 to conclude that M f : 1/)~l provided 0 < [IflIw_l,p,(u2) ~ (r Hence, u0 is another minimizer for //~f - - ~,1 in (6.19) which is not co-linear to ~01. D To summarize our results from this paragraph, we have shown that even if//~f --- )~1 holds for 0 < IIf [Iw-l,p' (s?) <<-(c/)~ 1) 1/p, there are two eigenfunctions ~01 and u0 associated w i t h ~ f which are not co-linear. This nonuniqueness (as opposed to the uniqueness in Corollary 3.5) is due to the fact that the arguments with u + and u - presented for ~,1 after formula (3.1) can no longer be applied to//~f in (6.19).

7. A saddle point method for p < 2

Similarly as in the previous section, for the sake of simplicity also in this section we restrict ourselves to the case F(x, u) = f ( x ) u , i.e., f ( x , u) -- f ( x ) is independent from u 6 R. Hence, the functional ff)~ introduced in (1.4) takes the form (7.1) P

P

437

Nonlinear spectral problems

for u ~ W~ 'p (S-2). In contrast to the case p > 2 in Section 6, Remark 6.6 (and under similar assumptions), for 1 < p < 2 the functional Jz~ will turn out to be unbounded from below on W~ 'p (f2) along curves "close" to +r~Ol as r --~ +cx~, even though it still remains coercive on the orthogonal complement Wo 'p (~2) T of lin{qgl } in W~ 'p (E2),

wl'P(ff2) -Vdef{. E w l ' P ( f f 2 ) 9 (U, qgl)=O}.

(7.2)

Hence, we take advantage of the orthogonal decomposition Wo'P(I-2)- lin{q)l} 9 w~'P(I2) T defined in (6.1) again. This picture shows that the functional JZl has a simple "saddle point" geometry. Such a scenario is typically suitable for a saddle point theorem ([51], Theorem 4.6, p. 24) which guarantees the existence of a critical point for ,.gzt by means of a minimax formula for a critical value of ,.7~. This observation was used in the work of Drfibek and Holubovfi [24], Theorem 1.1, to establish an existence and nonexistence result for problem (6.12) when 1 < p < 2. In this section we present their method.

7.1. Simple saddle point geometry The following notion is crucial. DEFINITION 7.1. We say that a continuous functional s

Wo'P(~-2) + ~ has a simple

saddle point geometry if we can find u, v ~ W~ 'p ( ~ ) such that (v, ~ol) < 0 < (u, ~ol) and max{s

,5'(v) } <

inf

s

wcW~'P(s2)T Notice that on any continuous path 0 " [ - 1 , 1]--+ wI'p(I2)

with 0 ( 1 ) - u

and

0(-1)-v there is a point w - O(to) ~ w0'P(12) T for some to 6 [ - 1 , 1]. Hence, max{g(u), g(v)} < g(w) shows that the function g o 0 : [ - 1 , 1] --+ R attains its maximum at some t' E ( - 1, 1). LEMMA 7.2 ([24], Lemma 2.1, p. 185). Let 1 < p < 2. Assume f ~ C ~ with (f, q)l) = 0 and f 7k 0 in s Then the functional ,.T),l has a simple saddle point geome-

try. Moreover, it is unbounded from below on W l'p (S-2). PROOF. We infer from Lemma 6.4 that A ~ > )v1 in formula (6.6). This shows that the functional JZl is coercive on C~ - Wl ' p (12)T. Hence, being also weakly lower semicontinuous, ,.7z! possesses a global minimizer u~- over Wl ' p (S'2)T, Jz, ( u ~ - ) -

inf W C WOI 'P (,.Q) T

J~, ( w ) > - o c .

438

P. Tak6(

Now let us look for the functions u and v, respectively, in Definition 7.1 in the forms of

u• = -+-z'qgl + Z'I-(p/2)r where r 6 C~ (s

with r 6 (0, cx~) sufficiently large,

(7.3)

is a function chosen as follows: First, recall our notation

U = {x ~ s

V~ol (x) # O}

and

U ' = 32 \ U = {x ~ 32" V~I ( x ) -

0}.

Then U' is a compact subset of S-2 with empty interior, by (5.3) combined with (5.13). Since f 6 C ~ satisfies f ~ 0 in s we must have f ~ 0 in U as well. In particular, U contains the closure of an open ball G C ~N such that either f > 0 in G, or else f < 0 in G. In either case we can easily find a function r 6 C~ (~2) that vanishes outside the ball G and satisfies (f, r 1. For r 6 (0, ~ ) we compute

(U-q-, ~l) = •

Ilgo11122
(7.4)

It follows that (u_, (/91) < 0 < (U+, qgl) for all r > 0 large enough. Next we use (5.9) and (5.10) to obtain

Oq'X1(/./-4-) : JA1 ('+-rqgl -+- TI-(p/2)q ~) = ~+r-p/2r162 r = ~•162

-- -cl-(p/2)(f, r

(q~, q~) -- r 1-(p/2).

(7.5)

We recall that the quadratic forms Q4_r-p/2r are given by formula (5.10). Since infG IV~011 > 0, infG ~Pl > 0, and r is supported in G by our choice of G and r we conclude that both summands in Q+r_p/2r (r r are bounded independently from r ~> r0, provided r0 6 (0, ~ ) is large enough. Finally, from (7.5) we deduce that ,.7~1(u+) ~ -cx~ as r --+ +cxz. The conclusion of the lemma follows. D

7.2. A Palais-Smale condition In order to be able to apply Rabinowitz's saddle point theorem [51 ], Theorem 4.6, p. 24, we need another lemma. LEMMA 7.3 ([24], Lemma 2.2, p. 188). Let 1 < p < 2. Assume f ~ W-I'P'(~f'2) with (f, ~Pl) # O. Then the functional Jz~ satisfies the Palais-Smale (P.-S.) condition, i.e.,

every sequence {Un}n~=l in Wo'P(12), such that Jz, (Un) --+ c ~ R and J ~ (Un) --+ 0 in W-I'P'(S2) as n --+ ~ , contains a strongly convergent subsequence in W~ 'p (12).

Nonlinear spectralproblems PROOF. Let {Un}nCc__1 be an arbitrary (R-S.) sequence in

that it is bounded in W1' p (s by {Un}n~=l, satisfies

439

W; 'p (s

As usual, we first show

On the contrary, suppose that a subsequence, denoted again ~ as n --+ ec. The R-S. condition implies

IlUnllwd,~(S2)~

&l

(bin) --- --

Pl f x 2

IV//n

[Un[p dx - fs2 f (x)un dx --+ c ]P dx -- -~'1 P fs2

(7.6)

and --1

f. where Vn - un/llu/7

Ilw~,p(x~)" We combine these two facts to get

'

w~,p(s?)

( 1 ) 1 -

(7.7)

f (X)Vn dx --+ 0

(Jr, (Un) Vn)- IlUnl1-1

-

(fs2 lVunlP dx _ Zl fs2 lunlP d x )

&l ("/7)

p-1 (f~ IlunIIw01,~(~)

P

]VvnlPdx-Xlfs2lvnlPdx)---->O.

It follows that Ilvnllwl,p(s2) - 1 and Ilvnllgp(x2) ~

(7.8)

~,7 lip. NOW we can argue similarly as

in the proof of Lemma 6.4 to conclude that Vn ~ -4-i,1 1/Pq)l h o l d s strongly in wI'P(I2) as n -+ ec for a suitable subsequence. Applying this result and (7.8) to (7.7) we arrive at

fs? f vn dx -+ -]'-XT1/P fI 2 f ~o] dx = O, a contradiction to our assumption (f, @1) ~ 0. Thus, we have proved that {Un}L1 must be bounded in W0 ' p (s Next, W l'p ( ~ ) being reflexive, we extract a weakly convergent subsequence, denoted again by {Un}n~__l,i.e., Un --~ u weakly in W d'p (s

as n --+ cx~. Hence, Un -+ u strongly in

L p (if2) by Rellich's theorem. From the definition of a R-S. sequence in Wo 'p (s we have (,.7~1 (u/7), Un - u) -+ 0 as n --+ ec. Using u, --+ u strongly in L p (S2) we observe that the last limit is equivalent to

P dx - fs? IVun IP-2VUn 9 Vu dx ~ O. We apply Young's inequality to the second integral to get lim sup Ilu/7 IIp

n-~ r

W(~'P(s )

~< l i m i n f ( 1 IlUn IIp

1

)

440

P Takd(

which entails limsup IlUn llw~,p(s?) <~ IlUllw~,p(s?). tl----~O0

On the other hand, Un ~ u weakly in Wd 'p (/2) yields Ilullw~'P (~) ~ liminf llun llw~'P

From the last two inequalities we deduce n~cclimIlun !1w0,~(s~)' - I l u IIw~,~(~) /

which, when combined with The lemma is proved.

Un __.x U again,

7.3. Fredholm alternative at

~,l

guarantees

Un

~ U strongly in Wo 'p ( ~ ) . U

Now we are ready to apply the saddle point theorem to the energy functional JZl defined in (7.1) in order to establish the following existence result for problem (6.12). This result complements Theorem 6.7, not only because 1 < p < 2, but also (f, ~01) 7~ 0. PROPOSITION 7.4 ([24], Proposition 2.1, p. 189). Let 1 < p < 2. Assume gT ~ c o ( ~ ) with (gY, q)l) = 0 and gY ~ 0 in S2. Then there exists a constant p =_ p(gY) > 0 such that, f o r any f E W-I'P'(S2) with (f, ~ol) 7~ 0 and Ilf - gT ilw_l,p, (s?) < P, problem (6.12) has at least one weak solution. PROOF. Since we keep ) ~ - ~,1 constant, but vary the function f ~ L~([2) in problem (6.12), it will be convenient for us to use the notation g f ( u ) de__Bfj.l(U; f) for u ~_ W; 'p (S2); cf. (7.1). By Lemma 7.2, the functional ggZ has a simple saddle point geometry. But this property clearly remains preserved for g f for all sufficiently small perturbations of gT, that is, also for f ~ W-I,P'(s with [ I f - gTIIw-l,p'([2) < P" Here, p _= p(gT) > 0 is a sufficiently

small number. Indeed, notice that fn ~ gT in W-I'P'(~Q) as g / ~ oo implies inf gf, ~eWo'e (n)S

(w) ~

inf ggZ (w), ~ewl'P (n)s

by arguments used in the proof of Lemma 6.4. According to Lemma 7.3 the functional s satisfies the R-S. condition for any f W-I'P'(I2) with (f, q)l) ~ 0. Next we take such f with IIf - gT IIw-~,/(s2) < P- Hence, by a standard variational argument (a saddle point theorem [51 ], Theorem 4.6, p. 24), the

Nonlinear spectral problems

441

functional ~f has at least one critical point which corresponds to a weak solution of the original problem (6.12). [3 To cover also the case ( = 0, excluded in Proposition 7.4, yet another method was applied in [24], Section 2, pp. 189-193, based on well-ordered and unordered pairs of sub- and supersolutions for problem (6.12). We will present this method in detail in Section 10. Therefore, here we only state the main result from [24], Theorem 1.1, p. 184; its proof will be given in Section 10. However, part (i) is a special case of Corollary 8.15 which will be established already in Section 8.6. THEOREM 7.5. Let 1 < p < 2. Assume f T 9 c o ( ~ ) ) with ( f T , qgl) - - 0 and f T ~ in s Then there exist two numbers ( , = ( , ( f T ) and (* =-- ( , ( f T ) with -cx) < ( , < 0 (* < oc, such thatproblem (6.12) with f - f T + (q)l has (i) no solution f o r ( 9 IR \ [(,, (*]; (ii) at least one solution f o r ( 9 [(,, (*]. Moreover, given any gT 9 c o ( ~ ) with (gT, q)l) - - 0 and gT ~ 0 in I2, there exists number p =_ p(gT) > 0 such that problem (6.12) has at least one solution whenever f L~ satisfies IIf - gTIIL~(S2) < P-

0 <

a 9

In Section 10 we will establish a much stronger result for any p 7~ 2, namely, that there are also two other numbers (# and (# with (, ~< (# < 0 < (# ~< (*, such that problem (6.12) with f - f r _+_(qgl has at least two distinct solutions provided (# < ( < (# and ( 7~ 0; cf. [58], Theorems 3.1 and 3.5.

8. Asymptotic behavior of large solutions A priori estimates play a crucial role in establishing existence results for various types of ordinary and partial differential equations and their systems. While deriving an a priori estimate, one usually attempts to estimate a suitable norm of an arbitrary solution (or an approximation thereof) directly. In [57], Section 5, pp. 206-215, a somewhat different approach to deriving a priori estimates has been introduced for problem (1.7) where the spectral parameter )~ 9 IR takes values near the first eigenvalue )~1 of - A p . This approach is based on a very thorough investigation of the asymptotic behavior of an unbounded sequence of possible large solutions u -- Un -- ux~+un 9 W; 'p (I2) of problem (1.7) with )~ = )~l +/Zn ~< )~2 - 6 (n = 1, 2 . . . . ) as n -+ ec. (Of course, 6 > 0 is an arbitrarily small number.) Recall from Section 6.3 that )~2 stands for the second eigenvalue of the positive Dirichlet p-Laplacian --Ap and there is no other eigenvalue in the open interval 0~l, )~2), by [3], Remarques 2.2, pp. 15-16. In particular, we will see soon that/Zn ~ 0 T and Un - t21 (q)l + VnT ) must hold with tn -+ 0 (tn 7~ 0) and IlVn IIc~,~' (~) --+ 0 as n ~ ~ . We view t = tn =/=0 as an independent bifurcation parameter and look for possible triples (t, #, v T) - (tn, #n, VnT ) 9 R x R x WO 'p (I2) T near the bifurcation point (0, 0, 0) such that u = t - 1 (qgl -+- v T ) verifies (1.7) with )~ = )~1 nt-/z. Recall that the orthogonal complement W~) 'p (I2) T has been defined in (7.2). The investigation of the asymptotic behavior

442

P. T a k d (

of v n7- as n --> e~ was continued in [58], Proposition 6.1, p. 331 (see (5.2)), from which a stronger version of Theorem 7.5 was derived for any p ~ 2 ([58], Theorems 3.1 and 3.5). Finally, even more precise, higher-order asymptotic results were obtained recently in [23], Theorem 4.1. We present these asymptotic results in this section; they are summarized in Theorem 8.7 (Section 8.4). Here, also the number ~" = ~'n in f = f T + ffq)l is a parameter depending on t = tn. Of course, one may fix either/x or ~', or fix their interdependence, in general. Finally, if the asymptotic dependence of/Zn, fin o r OnT on tn as n --+ ec, obtained in the manner just described, can be excluded by a hypothesis imposed on/Zn, ~'n or UnT , then, by a contradiction argument, we cannot have large solutions of problem (1.7). Consequently, we obtain the boundedness of the solution set indirectly rather than from an a priori estimate directly.

8.1. An approximation scheme In this paragraph we investigate an approximation scheme for a weak solution to the Dirichlet boundary value problem (1.7) provided f E L~(S-2) satisfies f ~ 0. Among other things we compute the asymptotic behavior of large solutions. We emphasize that the orthogonality condition (f, ~01) = 0 is not required in this paragraph. We study the following sequence of Dirichlet boundary value problems for n = 1, 2 . . . . : --Apun

=

()~1 + lzn)lunlP-2un + fn(x)

in S2;

Un = 0

on 0s

(8.1)

We often take advantage of the weak formulation of problem (8.1): For each n 6 N and for all ~b 6 W; 'p (,(2),

y2 ]VUn ]p-2 (VUn, Vq~)dx = (~, + ..) f~ lu~lP-2u~Oax + f~ fncpdx.

(8.2)

Here, {/Zn}neC=l is a sequence of real numbers, {fn}n~_j are given functions from L ~ ( ~ ) , and {Un}~_l are corresponding weak solutions to problem (8.1) in w~'P(~) which are assumed to exist. We assume that these sequences satisfy the following hypotheses: (S 1) )~l + #n ~< )~2 - 8 for n = 1, 2 . . . . . where 0 < 6 < )~2 - )~l. ($2) fn converges to some function f in the weak-star topology on L~ i.e.,

fn --~ f in L oc(~2) as n --+ oo. We require f ~_ 0 in ~ . ($3) IlUnllw0,P(s?) ~ cx~ as n --+ cx~. We identify L ~ (s with the dual space of L l(S2) in a standard way by means of the inner product (.,.) from L2 (,Q). This duality induces the weak-star topology on L ~ (S2). Any closed bounded ball in L~(I-2) is weakly-star compact; the weak-star topology restricted to this ball is metrizable since L l (s is separable ([66], Chapter V, w1).

Nonlinearspectralproblems

443

By the regularity result in L e m m a 2.2 ([3], Th6orbme A.1, p. 96), hypothesis ($3) is equivalent to ($3') []UnllL~(Y2) --> OO as n -+ ~ . Furthermore, since 0 s is assumed to be of class C ],~, for some 0 < ot < 1, we can apply another regularity result, L e m m a 2.3 ([18], Theorem 2, p. 829, [45], Theorem 1, p. 1203, and [62], Theorem 1, p. 127), to conclude that Un e C l'~ ( ~ ) , for some f l e (0, ol). Finally, if {#n}n~=l is bounded also from below, say, (S1 ~) - ~ ~< )~1 + #~ (~< ~2 - 3) for n = 1, 2 . . . . . where 0 ~< ~ < cx~, then hypothesis ($3) is equivalent to ($3 n) IlUn ]lcl,~(~) -+ oo as n -+ oo. In what follows we often work with a chain of subsequences of {(#n, f n , Un)}n~=] by passing from the current one to the next. Nevertheless, we keep the index n unchanged with the understanding that no confusion may arise. We commence with the asymptotic behavior of the normalized sequence /~n de__f -1 Ilun IIL~(~)Un as n --+ oc. Observe that each fin satisfies II~n IIL~(S2) - 1 and

--Aptln -fin - - 0

()~1 -Jr- lZn)]fftn]P-2Un +

1-p liunllL~(~)fn(x)

in s

(8.3)

on 0s

Since 0s is assumed to be of class C 1'~, for some 0 < o~ < 1, we conclude that fin E C l't~ ( ~ ) , for some/3 e (0, or), and the sequence {fin}n~__l is bounded in C 1'~ ( ~ ) , by the regularity result mentioned above (Lemma 2.3). We allow 1 < p < oo. LEMMA 8.1. L e t fl' ~ (0, fl). We have #n --+ 0 a n d the s e q u e n c e {fin}n~__l contains a convergent s u b s e q u e n c e fin --+ tc~pl in c l ' # ' ( ~ )

as n --+ oo, where x E R is a constant,

eke. 11~OlliLt(S2) - 1. In particular, we have Un - tnl(~0| + VnT ), where {tn}n~__l is a se-

quence o f real n u m b e r s such that Ktn > 0 a n d tnUn ~ 89 in S2 f o r all n large enough; moreover, tn --+ 0 a n d v n-r --+ 0 in C1,3 ' ( ~ ) as n --+ oo, with (vTn , q91) -- Of o r n -- 1, 2 . . . . .

This lemma generalizes [57], L e m m a 5.1, p. 207, where/Z n = 0 is assumed for all n e I~l; recall l~l = { 1, 2, 3 . . . . }. PROOF OF LEMMA 8.1. First, we show that the sequence {//~n}n~176 1 is bounded also from below. Let us take 4~ = Un in (8.2):

f~2 IVunOPdx -- (~'I -Jr-I~n) fy2 lunlP dx "+"f~ fn un dx" We apply the standard Poincar6 inequality (cf. (1.8) for ~,1 > 0) to the integral on the left and the H61der inequality to the second integral on the right to obtain

I ,i

+

i .l

+ HI/.

il .

(8.4)

444

P Takdg

By hypotheses ($2) and ($3), respectively, the sequence Ilfn IIL~(S2> is bounded whereas Ilunllw~,~ bounded and IlunllLp(m ~ as n - - + cx~, by hypothesis (S1). Hence, we p-1 --lZn Ilun IILP~S~) must be bounded from above for In particular, the sequence {/Zn}n~=l is bounded. Consequently, in the rest of this proof we #n --+ #* as n --+ cx~. We have 0 ~
deduce from (8.4) that the sequence all n e N. This forces l i m i n f n ~ / Z n

~> 0.

may extract a convergent subsequence X1 - 6 for every n = 1, 2 , . . . . Now we to the sequence {fin}n~_j to obtain another

convergent subsequence fin ~ tb in C 1 ' ~ ' ( ~ ) as n --+ cx~. Letting n --+ cx~ in the weak formulation of problem (8.3), we arrive at - - A p t o = (~.1 + ~*)ll~l p - 2 ~

in I2;

tb = 0

on OS2.

(8.5)

Since X1 is the only eigenvalue of --Ap in the open interval ( - o o , X2), we get/z* = 0. The eigenvalue )~1 being simple (Corollary 3.5), we conclude that tb = Xq)l in I2, where x e IR is a constant, tc r 0 by l]tb IIco~ (s2) = 1. Let {lZnk }~=1 be a subsequence of {/Zn}~-i such that [/zn~ I >/r/for some r / > 0. Applying the same argument as above we get a contradiction by obtaining a subsequence of {/Znk }~-1 that converges to zero. So, indeed, the s e q u e n c e / z n itself converges to 0, and not just a subsequence of it. The remaining statements are deduced from the identity

fn - t~ =

T 1 - tc Ilgo~ + On IIL~(s2) II~01 + v~ IIL~(s2)

qgl +

T On II~ol + v~ IIL~(s2)

We combine [IvnT Ilcl,~, (~) -~ 0 with the Hopf m a x i m u m principle (5.13) for 991 to find out that Iv nT I ~< 89 in f2 provided n is sufficiently large, say, n ~> no. In particular, we get q91 -+- 1)n ~ lq91 > 0 in I-2 for every n ~> no.

[--]

As a consequence of L e m m a 8.1, for each n = 1, 2 . . . . . we can rewrite problem (8.3) as +

--(•1 VnV - O

T p-2 +//~n)[qgl + On ] ( qgl -+- vTn) -+- ItnlP-2tnfn(X)

in g2,

(8.6)

on 0S-2

with all tn =/: O, tn --+ 0 as n --+ oo. Furthermore, if x < 0, we can take advantage of the (p - 1)-homogeneity of problem (8.1) and replace all functions fn, f and Un by - f n , - f and -Un, respectively, thus switching to the case x > 0. Hence, without loss of generality and whenever convenient, we may assume tn > 0 and tnUn -" 991 -+- VnT ) 1991 > 0 in s for all n >~ 1, with tn "N 0 as n --+ cx~.

Nonlinear spectral problems

445

8.2. Convergence of approximate solutions A very useful equivalent form of problem (8.1) is the following one obtained by subtracting (5.3) from (8.6) and using the integral Taylor formula with a help from identity (5.5): - div(A,Vv:) -- (p -- 1)(X1 + lZn)anV nT + #ntpl~ 1 + Itn ]p-2tn in (x) Ur7/ m 0 on 0s

in S2,

(8.7)

T

with the abbreviations

-fo

An dee

1A(Vqgl +sVvTn)ds

and

def/o'

an -

T p-2

]~P1 + SVn ]

as.

(8.8)

Recall that the matrix A(a) is defined in (5.6). We abbreviate also A~ol def A(V~Ol)

and write

(8.9)

A 1/2 -- ~//A{/91 --991

This means that each function Vn def (it~lP_2tn)_ 1v nq- 6 ear boundary value problem -

cl,fl,

( ~ ) (n ~> 1) satisfies the lin-

- div(An V V~) = (p - 1) (~ ~ + l*~)a~ y~ +

#n

~o~-I + f~ (x)

[tn[P-Ztn

V~ = 0

in s (8.10)

on 0s

(v~, ~o~)=0. In order to determine the limits of Vn and lZn/(ltnlp-2tn) as n -+ oo, we need the following two "universal lemmas" for p > 2 and 1 < p < 2, respectively. We keep our Hypothesis (HI) for any 1 < p < oo and (H2) for p > 2 throughout the remaining part of the present section. Recall that (H2) holds always true for 1 < p < 2; see Section 5.5. LEMMA 8.2. Let 2 < p < cx~. Assume that 0 < Otn < O0 and v nT ~ W o1,p (s N C 1 ( ~ ) satisfy v nT - - a n Vn ---->0 strongly in C 1( ~ ) , and Vn ~ V weakly in L2(~2) as n ---->cx~. In addition, assume that Rn ~ R weakly in L 2 ( ~ ) and

fs2(

A n V V n , Vck) dx -- j's2 R n C d x

f o r a l l C e w~'P(s2).

(8.11)

A1/2W Then also V E 79~9~, which implies "'r - V ~ [L2(~(2)] N, and

fs2 (Ar VV, Vr dx - fs2 R e dx

f o r all r ~ 7Pr

(8.12)

446

P Tak6(

Moreover, we have Vn --+ V strongly in Dr [L2(~Q)] N as well.

and A1/2 ,-n VVn --+ A1/2V/ "~r

strongly in

A complete proof of this lemma, based on inequalities (5.37) and (5.42), is quite technical and is given in [23], Lemma B.1. It is derived from the proofs of Lemmas 5.3 and 5.4 in [57], pp. 210-213. This lemma will be needed several times later, with a more general fight-hand side in (8.10). For 1 < p < 2 we need to employ "improper" integrals of type An and an defined I in (8.8). In analogy with D~01 being the dual space of D ~ (cf. Section 6.2), here we denote by 7-/~j the dual space of 7-/~0~ (cf. Section 5.5), with the duality induced by the inner product (.,.) in L 2 ( ~ ) . Recall that the Hilbert space 7-/~o~is continuously embedded in L2(I2); I see Lemma 5.8(a). Hence, also the embedding L2(f2) ~ 7-/~0 ~ is continuous. LEMMA 8.3. Let 1 < p < 2. Assume that 0 < Oln < (X) and 13nT C_ W 01,p (S-g) (] C 1( ~ ) satisfy 13nT "--olnVn ~ 0 strongly in CI(,Q), and Vn ....x V weakly in ~]'[q91 as n --+ oo. In

! addition, assume that Rn ~ R weakly in "H~o1 and

ff2(

AnVWn, Vq~)dx -- fl2 Rnqbdx

for all q~ E W0'2(S'2).

(8.13)

A1/2V7 Then also V E 79~Ol,which implies --~o~ - V E [L2(~(2)] N, and (8.14)

is2 (AqglV V , Vqb) dx - fs2 R $ dx f o r all ~ E ~)q)l. Moreover, we have Vn --+ V strongly in Wo'Z(f2) and A 88

A1/2V/ --+ "~1 -V

strongly in

[LZ(~Q)] N as well. Again, a complete proof of this lemma, based on (5.38) and (5.43), is given in [23], Lemma B.2. It is derived from the proofs of Lemmas 8.4 and 8.5 in [57], pp. 227-228. REMARK 8.4. Although we have formulated the auxiliary results in Lemmas 8.2 and 8.3 for An only, analogous claims remain valid (with the same proofs) also for A(n2) def --

1 fo A(Vqgl + s V v nT )(1 - s)ds.

(8.15)

8.3. First-order estimates In this and the next paragraphs we present the asymptotic formulas obtained recently in [23], Section 4. They improve an earlier result from [58], Proposition 6.1, p. 331. From T IIc,,~' (~) ~ 0 and/Zn ~ 0 as n ~ oo. Next Lemma 8.1 we know that tn ~ 0 implies IIVn we compute the limits of Vn and lZn/(ltnlP-2tn) as n --+ oo.

Nonlinear spectral problems

447

PROPOSITION 8.5. Let 1 < p < ~ , p ~: 2, and let {#n}n~=l C ~, {fn}n~__l C L~(Y2), and

{Un}n~__l C W I'p (Y2) be sequences satisfying hypotheses (S1), ($2) and ($3), respectively. In addition, assume that they satisfy (8.2)for all r 9 W~ 'p (Y2) and for each n 9 N. Then, T writing Un -- tnl (Cpl + v n) with tn 9 R, tn ~: O, and v nT 9 W o1,p (Y2 ) T , we have tn --+ O as n ---> ~ , Vn _ (itnlP-Ztn)-I VnT __+ V T strongly in 79~ol if p > 2 and in wl'Z(Y2) if 1 < p < 2, and

lim /~n -- - f fcPl dx. n-+~ ItnlP-2tn ds2

(8.16)

Moreover, the limit function V y 9 79~ N {(/91}I'L2 is the (unique) solution to

2. Q0(V T, r

-- fs-2 f t r dx

(8.17)

for all r 9 79~,,

where the symmetric bilinear form Qo is given by (5.11)and f t _ f _ ( fs2 f qgl dx)~o~-1.

Formula (8.16) provides an asymptotic estimate for /z n of the first-order relative to a s tn ---> O, i.e.,

]In ]P-2tn

(8.18)

lZn -- --ItnlP-2tn fs2 f go1 dx + o(ItnlP-1).

We will improve this estimate to a second-order one in the next paragraph. REMARK 8.6. The linear equation (8.17) represents the weak form of the "limiting" Dirichlet boundary value problem for the limit function Vn - (itnlP-2tn)-I VnT __+ V T in the approximation scheme with Un = tn 1(qgl + vT). This is a resonant problem to which a standard version of the Fredholm alternative for a selfadjoint linear operator in a Hilbert space applies. More precisely, given a function f 9 L 2 (#2), a weak solution V 9 79~0~ to the equation 2. Qo(V, r

- fs~ f r dx

for all r E D~0~,

(8.19)

exists in D~01 if and only if fee fqgl dx - 0. Such a solution is always unique under the orthogonality condition fs2 Vqgl dx - 0. Formally, (8.19) is equivalent to the following linear degenerate boundary value problem obtained by linearizing (1.7) with )~ - - ~.1 about r - div(A(Vcpl)VV) - ~l (P - 1)r V-0

+ f(x)

in Y2,

(8.20)

on OS2.

We stress that the observations just made remain valid also for 1 < p < 2 when the corresponding selfadjoint linear operator has to be considered in the Hilbert space ~L2 the (/91 '

448

P Tal~(

closure of 79~ol in L 2 ( ~ ) ; see Section 5.2. Then, of course, only the orthogonal projection ~C 2 of f to D~01 matters in (8.19), according to the orthogonal sum

L 2 ( ~ ) -- ~ L2 7)/, L2 (/91 (~ q?l " Consequently, given f-r E {qgl}•

(8.21) C L 2 ($'2), we denote by

V 7- ~ V-I-(fT) E 79~0, n {(/91}• the unique weak solution to problem (8.19) with fq- in place of f . It is easy to see that fq- w-~ V T" {qgl}• --+ D~0, is a compact linear mapping. Clearly, this mapping is linear and bounded. To show that it is compact, let {fn }nCr C {qgl}• be any weakly convergent sequence, fn --~ f in L2(s as n ~ oo. Hence, {V T(fn)}n~__l is a weakly convergent sequence as well, v T ( f n ) --~ v T ( f ) in 79~ol as n --+ oo. The embedding 79~01 ~ L2(s being compact, we have also V T (fn) --* V ~-( f ) strongly in L2([2), and

f

fnCdx ~ f ~ fc~dx

uniformly for 4) 6 79~ol with [lr

~< 1. Inserting these results into (8.19) we deduce

~ (A~olVVT(fn) , V~b) dx ---> f ( A g l

vvT(f),

V~b)dx

uniformly for 4) 6 79~01 with II~ll~l ~< 1. We have shown v T ( f n ) -+ v T ( f ) in 79~01,and thus the desired compactness.

strongly

PROOF OF PROPOSITION 8.5. We have already shown in L e m m a 8.1 that tn 5~ O, tn ~ O, ~. /An ---> 0, and IIvn7- IIc, ,/v (~) ~ 0 a s n ~

Step 1. We now claim that VnT def_([tn[P_etn)_ 1VnT is a bounded sequence in Le([2) if p > 2 and in ~ if 1 < p < 2, and that/An/[tn[ p-1 is a bounded sequence in I~ as well. By contradiction, let us suppose that this is not the case. We set I for n = 1 . .2, Nn de=f 1[gnT II + [tnI#n [P---------~ ...

where

IIV~ll

denotes either

IIV~IIL2(X~) if

p > 2 or

(8.22)

IIVnWll~

if 1 < p < 2. Thus, we

may assume without loss of generality that Nn --+ (x). We set WnT de_...fgnT/Nn" Then ][WnT[[L2(S2) ~ 1 if p > 2 and IIW~117%~ ~< 1 if 1 < p < 2. In addition, from (8.10) we

Nonlinear spectral problems

449

obtain the corresponding equation for Wn-r, f ~ , ( A n V W f , V~b)dx

= ~,1 (P -- 1) fs~

+

NnitniP-2tn

an -T- ^p-2 p-2 w~ ~l 4) dx

f ,f_,Cdx + # n ( P -

1)

L

'L

anWTndpdx + -~n

fndpdx

for all ~b E Wo '2 (ar2). Set

Rn de__f)~l (P

..,-T- p-2 -- 1) an p-2 wn ~~

]Zn

p-1

+ Nn [tn iP-Ztn ~01

1

+ I~n(P -- 1)anWYn -+- -~nfn. We consider the case 1 < p < 2 first. There exist constants Cl > 0 and C2 > 0 such that, for every n sufficiently large, we have

Cl ~ an(X)99f-2(x) ~ c2 and moreover,

an/q)f -2

for all x ~ $2,

----> 1 as n --+ c~ uniformly in s

Since IlWnq- I ] ~ l ~ 1, it follows

IlanWTnllT-t~0~ ~< c2. Passing to a subsequence if necessary we may assume Wn-r ~ W T an WnT ~ 9ff -2 W T weakly in 7-t~ for some W T E 7-t~. Note that fn/Nn --* 0 strongly in L ~ ( s and I~n/(NnltnlP-2tn) --> 0 with some 0 E [0, 1]. Then Rn --~ R weakly in ~ l ' where

that

weakly in ~0~ and

R de__f~,l (P -- 1)qgf - 2 W T + 0~0f -l" Now let us consider p > 2. Since IIWnT [ILZ(n) ~< 1 andan ~ qgf -2 as n --> oo uniformly in s passing to a subsequence if necessary, we deduce that Wn-r ~ W T and Rn ~ R weakly in L 2 (at-)). By Lemmas 8.2 and 8.3, there exists a subsequence of {WnT}n~=l such that Wn-r ---> W ~strongly in Z)~0~ if p > 2, in W1'2(s

if 1 < p < 2, and W ~- E D~01 satisfies the equation

fs2(A~i VWq-, Vdp)dx - )~l (p - 1 ) fs q)p-2wq-dpdx + Ofs ~op-1 ~bdx for every ~b E D~I. Taking ~b - tpl in (8.23) we get

(8.23)

450

P

Tak6(

The left-hand side of this equation equals to 2. Q0(~01, W T) = (.A991go1, W T) = 0 and thus yields 0 = 0. But this and taking ~b - W T in (8.23) show that Q0(W T, W T) = 0, and thus W T - x q)l for some constant x ~ R; see Proposition 5.4 if p > 2 and Remark 5.7 if 1 < p < 2. Due to fs2 WTq 91 dx = 0 we have W T - 0 . Summarizing these convergence results, we find Wnv = VnV/Nn ~ 0 strongly in L2(s if p > 2, in 7-/~ol if 1 < p < 2, and lZn/(Nnltnl p-l) --+ O. Therefore, 1-

N~

_

IIVnTII

+

(Itnlp-1)-lllznl ~ 0 as n ~ cx~

N,

N.

which is a contradiction. We have verified that both IlVnT II and lZn/ltnl p-l are bounded. T [p -2tn) --+ V T strongly in D ~ if Step 2. Now we prove (8.16) together with vn/(Itn p > 2 and in W~ '2 (s if 1 < p < 2. We make use of similar arguments as in Step 1. Again, from (8.10) we deduce

f (AnVVT, Vr~)dx =)~l (P -

1)fnanV~g~ctr

4-itn ip_2~ (8.24) for all 4~ 6 w l ' 2 ( f 2 ) 9 Since the sequence {Nn}n~=l defined in (8.22) is bounded, by Step 1, we may assume (by passing to a subsequence if necessary) that VnT ~ V T weakly in L2(f2) if p > 2 (in 7-/~o~ if 1 < p < 2, respectively) and lZn/([tn[P-Ztn) ~ 0 for some 0 6 R. Now we apply Lemma 8.2 (8.3, respectively), with a new Rn, namely,

Rn de__f~.I(P -- 1)an V.T 4-

~n qgf- I + lZn(p- 1)an Vn7 + fn. [tn[P-2tn

The computations above imply that

gn ___xg de_.f~,1 (P -weakly in L 2 ( ~ )

1)cP~'-2V-T-+

(in 7-/~01,respectively). Therefore, the limit equation reads as follows:

s_2(A(pl v v T ,

V~b) dx - ~,I(P - 1) f n ~Of-2vTr

+0 for all 4~ 6 Del, and in W~ '2 (s

O~of -1 ' [ f

~o1

~dx+

cL~ f4~dx

(8.25)

vnT I(Itn IP-2tn) --+ V v strongly in 79~ if p > 2 (by Lemma 8.2) and

if 1 < p < 2 (by L e m m a 8.3).

Nonlinear spectral problems

451

In particular, for r = q91 we get

fs2(

A o, V 0,, v V T ) d x -

(p - 1) fs2 qgf-' V T dx

that is, 0 - 0 + fs2 f~01 dx, by fs2 qg~ dx - 1. This proves (8.16). Using 0 - - fs2 fqgl dx and defining f t

-

-

f - (fs2 fqgl dx)q9p-1 , we can rewrite (8.25) as follows"

fs2(A~oi v v T , v r

- )~l(p - 1 ) fsecp~-2vTCdx = f ~ f t C d x

for all r 6 79~ol,which is (8.17). The orthogonality condition fs2 VT qgl dx - 0 follows from the fact that fs2 VnT qgl dx = 0 for all n ~ 1~. Note that, by Remark 8.6, there is precisely one function V T satisfying (8.17). Thus, the strong convergence of the whole sequence T vn/(ItnlP-Ztn) ~ V T follows by the standard argument used towards the end of the proof of Lemma 8.1. The proof of the claim of Step 2 and of the entire proposition is now finished. D

8.4. Second-order estimates The following improvement of Proposition 8.5 is due to [23], Theorem 4.1. Its onedimensional "relatives" (but not analogues), for #2 = (0, a) with 0 < a < ~ , can be found in [48], Eq. (2.5), p. 393, for f 6 cl[0, a] and in [47], Eq. (46), p. 335, for f 6 Ll(0, a) and at any eigenvalue )~k (k ~> 1). THEOREM 8.7 ([23], Theorem 4.1). In the situation of Proposition 8.5 and under the same hypotheses, the asymptotic formula (8.18) has the following improvement as tn --+ 0:

lZn -- --It, lP-2tn iS2 fng01 dx + (p - 2)ltnl 2(p-1) Q0(V T, V T)

+(p-l) It.12(p-1)(ff ,dx)(f P-'VT dx) +

(8.26)

In particular, if fs2 fn 991dx = 0 for all n e I~, then #n -- (P - 2)Itn]Z(P-1)Qo(VT, V T)

-~-O(Itnl2(p-1)).

(8.27)

452

P Tak6(

On the other hand, if lZn = 0 for all n ~ N, then 1

lim n-+oo Itn IP-2tn

f n q)l dx

- < p - 2~Qo(V~,

v~)+~p-l~(f~f~oldx)(fs2~oP-1

VT dx).

(8.28)

It is now quite clear how to obtain "indirect" a priori estimates for weak solutions of problem (1.7) provided )~ takes values near )~l, p #- 2, and

f* = f

-

f,Q f g o l dx t qgf- 1 ~ ~)q91 I'L2

in s

(8.29)

For the (unique) solution V T E D,~ n {(p1}I'L2 to (8.17), condition (8.29) entails V T r 0 in I2 and therefore also Q0(V T, V T) > 0, by Proposition 5.4 if p > 2 and Remark 5.7 if 1 < p < 2. Then, for instance, if/Ln = 0 for all n 6 N, formula (8.27) leads to a contradiction. In other words, the sequence {Un}n~=l must be bounded in C I ' ~ ( ~ ) , by hypothesis ($3") which is equivalent to ($3). We postpone the details until the next subsection (Section 8.5). PROOF OF THEOREM 8.7. We take the inner product of (8.6) with 4~ - ~ol + v nT to get

IV~I +VUnTI pdx-,~l

I~ 1 + Vn ] dx

Iqgl + 1)nTIP dx --1-[tn]P-2tn ~ fn((Dl--]-u~)dx.

= ].Ln

Next we apply (5.9) and (5.10) to obtain p

f<[fo

]

-p(p-1)akl

--lZn

)

A(Vqg, + sVVnT)(1 -- S)as VvnT, Vvn7 dx

(fo

~f dx + p

If0'I~Ol+ s v n9I P - 2 ( 1 - s ) d s ] (vnY)2dx

fo If0'

] )

I~01 + SOnv Ip-2 (~01-'[-SvV) as Onv ~

+ltn[P-2tn~fngOldX+ltn[2(p-l)~fnV~dx.

(8.30)

Let us recall the abbreviations An, an, and A~2) introduced in (8.8) and (8.15), and introduce also an(1) deff01 = 199, + SVnT Ip-2 (9' + SVTn )ds.

(8.31)

453

Nonlinear spectral problems

Also note that (A~2)v, v) ~< (Any, v) for all v E ]I~N pointwise in £2. Dividing (8.30) by Itn 12(p-2) a n d using v nT = ItnlP-2tnV~ we arrive at p f£2 (A~,2)VV~, VVnT}dx -- p ( p -- l)~,l f,Q ~n t Vn ! dx "(2)/~1T~2

-

1 ( ItnlP-Ztn

[Zn ItniP-2tn

'/o

+ It,,IP 2t~

~0p dx + p

f~oldx+

£~('),,~dx) t~n v n

foLV~Vdx.

(8.32)

Set t--(2)~ Qn(V, W) d.eef -- Jsf tan VV, g w ) d x - - ~ . I ( P - 1 ) f s 2 a ( 2 ) v w d x , and recall that the symmetric bilinear form Q0(v, w) is given by (5.11). We wish to pass to the limit as n --+ cx~ in (8.32). We have Vff --+ V T strongly in D~t ~ L2(£2) if p > 2 (in Wd'2($2) ~-+ L 2 ( ~ ) if 1 < p < 2, respectively) and An112V V nT -~ A~2VV v strongly in [L2(f2)] N. If we pass to a subsequence (denoted again by {V~ }n~__l), we can assume also Vff --+ V T and An1/2VV~T --+ A1/2wwT -~Ol - - - pointwise a.e. in £2, and there are functions

hi, h2 E L 1(£2) such that Iv~(~)l ~ ~< hl(x )

and

I(A22))'12VVff(x)l 2 <~ IAU2VWf(x)l 2 <
for a.e. x c S2 (see, e.g., [44], Theorem 2.8.1, p. 74). Then, by the Lebesgue dominated convergence theorem,

Q,, (v. ~, v ; ) ~

Qo(V T, v ~)

.s ,, + oo.

Since fn - - f weakly-star in L~(~2) by hypothesis ($2), and a~ 1) --+ ~o~j - I strongly in L ~ ( s 2 ) , we have also f~a~l)v] ax -+ f ~ f - ' V T Hence, (8.32) yields

dx and f~ fnV] d~ -+ f~ i V T d~.

p . Qo(VT' v T ) -- is2 f V T dx = lim n--,~ ]tnlP-2tn +p

(f,,

]tniP-Ztn

~0('-lVTdx

~o~ dx +

) nlirn ItnlP-2tn"

fn~Ol dx

P. Takd(

454

Recall fs? q)~ dx = 1. Taking into account the limit (8.16), we arrive at p. Q0(vT' vT)

-- fs? f VT dx

'("" f. )

= lim

n~oo ItnlP-Ztn

-p(f.

ItnlP-Ztn

+

fnqgldx

f'l~) (f '~'-lw~)9

(8.33)

On the other hand, choose 4) = VnT in (8.24) to get

fs2(AnVVnT, VVnT)dx - ~I(P - 1 ) fsean(VnT)2 dx -- fse fnVnT dX

_--itnlp_2tn ~'" f ~f-' VV dx + ..(p _ ~) f an(VnV)2d~. We pass to the limit for n --+ cxz to get

f

(8.34)

Now we subtract (8.34) from (8.33), thus arriving at

+ lim

n-~c~ ItnlP-Ztn

~ + ItnlP-Ztn

fn qgl dx

which means

~n

L fngOldx

itnlP_2t n -+-

= [tnlP-2tn[(p - 2). Q0(V T, V v)

From this equation we finally derive (8.26). Due to the uniqueness of the limit, we use a standard argument to conclude that this asymptotic behavior holds for the original sequence {/Zn}n~__l as well. Theorem 8.7 is proved. D Theorem 8.7 has a very useful consequence for )~ =/~,1 and p r 2, namely, (5.2) established in [58], Proposition 6.1, p. 331, cf. (8.35). More precisely, for n -- 1, 2 . . . . we take

455

Nonlinear spectral problems

-2 /Z n - - 0 , f n -- f n T + (n(tgl where (n -11991 IIL 2 ( ~ ) f ~ fn(fll dx, and instead of (S2) a s s u m e

only ($2 -r) f ~ converges to some function f-r in the weak-star topology on Le~(I-2), i.e., __..x

-1-,L2

fn-r * f-r- in L~(S2) as n --+ cxz. We require f T ~ 79~o1 . Hence, the sequence {(n }n~=l C R is not assumed to be a priori bounded. COROLLARY 8.8. In the situation of Proposition 8.5, with I~n -- 0 (n ~ 1) and with hypothesis ($2) replaced by (s2T), we have (n --> 0 as n --> oo, and moreover,

(n nlim ~ Itnlp-2------~,, = (p - 2)Ilqg111[2(52) 9 Q0(v

v, v v) r 0.

(8.35)

PROOF. Without loss of generality, we may assume tn > 0 for all n >/ 1 and tn --+ 0 as n --+ ~ . Indeed, if tn < 0 for an index n, we take advantage of the (p - 1)-homogeneity of problem (8.1) and replace the functions fn, f T , ~n and Un by - f n , _ f T , - ( n and -Un, respectively, thus switching to the case tn > 0. So tn > 0 and hence also tnUn - 991 -k- v nT >/ 1 qgl > 0 in $2 for all n ~> 1. By contradiction, suppose first that {(n}n~=l is unbounded. Keeping the same notation for a suitable subsequence, let I(nl--+ c~ as n --+ c~. For each n >/ 1, let us replace fn -- f~- + (nq91 and Un by fn de_f( n 1f~- + qgl and fin def i(nl_p/(p_l)(nUn ' respectively.

Consequently, each pair (fin, fn) satisfies (8.1) in place of (Un, fn), with #n - - 0 . Furthermore, we have Ilfn - ~0111L~(S2) --+ 0 as n ~ ~ . If the sequence {fin}n~__l contains a subsequence that is unbounded in L ~ (S2), we can apply Proposition 8.5 (formula (8.16)) with f - qgl in place of f to conclude that ( f , qgl) - - 0 , a contradiction. So {fin}n~__l is bounded in L ~ (~2), and consequently, also in C l'~ ( ~ ) for some/3 E (0, or), by regularity (Lemma 2.3). Fix any/3' E (0, 13) and invoke Arzel?a-Ascoli's theorem in order to pass to a convergent subsequence fin --+ fi in c l ' ~ ' ( ~ ) . Thus, letting n --+ c~ in the weak formulation of problem (8.1) with (fin, Jgn) and #n -- 0, we arrive at -Apfi=XllfilP-2[t+qgl(X)

in 1-2;

fi-0

on 0$2.

But this equation has no weak solution by the nonexistence result from Theorem 4.2. We have shown that {(n }n~- 1 is bounded. Now, again by contradiction, suppose that the sequence {(n}n~__l does not converge to zero. Hence, it must contain a convergent subsequence (n --+ ( E R \ {0} as n ~ cx~. It follows that fn ~ f in L ~ ( I 2 ) as n --+ ~ , where f def f-r + (qgl. But (f, ~01) -(11~0~112 c2(s~) ~ 0 contradicts Proposition 8.5 (formula (8.16)) again. We have verified *

m

(n ~ 0 as n -+ cx~. In particular, the sequence {fn}n~=l satisfies hypothesis ($2) with (f, ~ l ) -- O.

Finally, formula (8.35) follows directly from (8.28).

[3

The convergence in Theorem 8.7 above is uniform for fn ----f (n -- 1, 2 . . . . ) from any bounded set in L oc (s More precisely, we have the following corollary"

456

P. Takd(

COROLLARY 8.9 ([23], Corollary 4.4). Let K be a closed bounded ball in L ~ ( : 2 ) . Assume that fn = f (n = 1, 2 . . . . ) and tn --> 0 as n ---> cx~ in Theorem 8.7. Then there exists a sequence {r/n}neC__lC (0, 1), On ---> 0 as n -+ cx~, such that f o r all f ~ K and f o r all n = 1, 2 . . . . we have

],'--2(p--l)(/A,n--]tnlP--2/nL f991dx) -- (p -- 2) " o(V-F-,v T)

-(p-1)(f. f

(8.36)

o..

The sequence {JTn)L1 depends on K and {tn)n~176 , but neither on the choice o f f E K nor on the sequence {lZn }n~l C (-oo, 8t] where 8: = ~2 - )~1 - 8 > O.

PROOF. Assume the contrary to (8.36), that is, there exists a number )7 > 0 and a sequence {fn }n~=l C K such that, for all n = 1, 2 . . . . . we have

Itnl-2(p-1)(lzn-ltnlp-2tnfs2fngOldx )

- (p - 2 > e o ( V T(:+.),

--(p--1)(L

v T(:2))

fn~ldx)(Lfpf-lvT(f~)dx)

~/ YI.

(8.37)

Here, the function V T (f~) E D~01A{~01}-L'L2 stands for the weak solution to problem (8.17) with fn* in place of f t , where

fff -- fn -- ( f

fnflgldx)ftgf -1

and

ft--f --(f:2fqgldx)~f-1;

see Remark 8.6 above. The ball K being weakly-star compact in L ~ (~2), hence metrizable in the weak-star topology, from {fn }n~-i we can extract a weakly-star convergent subsequence fn *-~ f in L~(~2). We apply Theorem 8.7 to conclude that

[tnl-2(p-1)(~n--ltnlP-2tnfs 2 fn~Oldx) - (p - 2). Qo(V ~(:+1, v ~(:+))

--(P--i) (fs2

f~~176

v-T-(f?) dx)

0

(8.38)

Nonlinearspectralproblems

457

as n --+ oo. On the other hand, fn ~ f weakly-star in L ~176 (s yields fn ~ f weakly in L 2 ( ~ ) . From Remark 8.6 we infer V T ( f 2 ) --~ V-C(f t) strongly in D99~ for p > 2 (W1'2(~2) for 1 < p < 2, respectively). It follows that [~ 0 ( g T ( f n t ) ,

gT(fnt))-

~0(gT(ft),

vT(/*))I ~

0

(8.39)

and

(f~ fnqgldx)(fy2q3P-lgT(ftn)dx ) - (/~f~oldx)(/~qgp-lgT(f~f) dx)

0

(8.40)

as n --+ oo. Finally, we combine (8.38)-(8.40) to get a contradiction with inequality (8.37). The uniform convergence (8.36) is proved. []

8.5. A priori bounds We recall our Hypotheses (HI) for any 1 < p < oo and (H2) for p > 2. Proposition 8.5 and Theorem 8.7 have the following applications to a priori estimates which play a decisive role in obtaining our existence and multiplicity results in the next sections. First, let us consider the spectral problem (1.7) with - o o < X ~< )~2 -- 6, 6 > 0. THEOREM 8.10. Let 1 < p < cx~, p :~ 2, and let 0 < 6 < X2 -- )~1 and 0 <<,X < c~. As-

_L,L2

sume that K is a nonempty, weakly-star compact set in L~176 such that K N 79991 -- 0 and (f, ~Pl) - 0 for all f ~ K. Then there exists a constant C ( K ) > 0 with the following property: Any weak solution u E Wlo 'p (#2) to problem (1.7) obeys IlUilc~,~(~) ~< C ( K ) , provided f ~ K and X ~ R satisfies (a) -)~ 2 ; (b))~1 ~.~)~2-6

if p <2.

If the hypothesis X >~ -~. in (a) is dropped, then the estimate IlUllcl,~(~) ~< C ( K ) has to be weakened to IlUllw~,p(s~) + Ilullc~(#2) ~< C(K). This theorem is due to Takfi~ [57], Theorem 2.1, p. 194, for p > 2 and 0 ~< X ~< )~1, and to [57], Theorem 2.6, p. 196, for 1 < p < 2 and X - )~1. I,L 2 We recall from Section 5.2 that the orthogonal complement D99~ of 19991 in L 2 (#2) might be nontrivial if 1 < p < 2, i.e ", D q91 Z'L2 ~ {0} Of course, if p > 2 then the condition /,L 2 K N D99~ -- 0 trivializes to 0 r K. W e set 6 f - X2 - ~.1 - 6 > 0 a n d • - ~.1 -+-/~. On the contrary to the conclusion, suppose that there are three sequences {#n}nC~=l C

PROOF OF T H E O R E M 8 . 1 0 .

(-cx~,6'], {fn}n~__l C K, and {Un}n~__l C Wo'P(a"2), such that each X = X1 +/Zn obeys

458

P. Tak6(

(a) and (b), i.e., (p - 2)/Zn ~< 0, and Ilunllwl,p(n ) ~

~ as n ~ c~. The set K being

weakly-star compact in L ~ (~2), we may extract a weakly-star convergent subsequence

fn ~ f in L ~ (~2) as n ~ c~. Hence, f 6 K. We observe that now all three hypotheses (S1), ($2), and ($3) from Section 8.1 are satisfied. Recall that, under condition (SLY), ($3) is equivalent to ($3"): Ilun IIc~,~<~) --+ ~ as n ~ c~. But then we can apply Theorem 8.7, (8.27), to get 0 >~ ( p -

2 ) - l # n -Itnl2~P-~)Qo(W -w, V -w) -+-o(Itnl2(P-~)).

This forces Q0(V v, V T) ~< 0, whence V -r - - 0 in ~2, by Proposition 5.4 if p > 2 and _L,L2

Remark 5.7 if 1 < p < 2. From (8.17) and (8.29) we get f t _ f 6 D~0~ , a contradiction _L,L2

to our assumption K N D~0~

= 0. The theorem is proved.

[]

The following theorem complements Theorem 8.10; it was established in [23], Theorem 5.5. THEOREM 8.1 1. Let 1 < p < cxz, 0 < 6 < )~2 - ~1, and 0 <~~. < cx~. Assume that K is a nonempty, weakly-star compact set in L~($2) that satisfies (f, qgl) r Ofor every f ~ K. Then there exists a constant C (K) > 0 with the following property: Any weak solution u Wo'P (I2) toproblem (1.7)obeys Ilullcl.~(~) ~< C(K),provided f ~ K satisfies (f, qgl) > O, and )~ ~ • and u satisfy either of the following two conditions: (a) -)~ ~< )~ ~< ,ki and u (J ) <<.Ofor some 2c ~ ~2 ; ( b ) ) ~ 1 ~ )~ ~ ~,2 - ~ and u(J) ~ Ofor some ~ ~ s The corresponding result holds also for f ~ K satisfying (f, qgl) < 0, with the reversed inequalities for u(~) in conditions (a) and (b) as well. If the hypothesis )~ ~ -)~ in (a) is dropped, then the estimate IlUllc~.~(~) ~< C ( K ) has to be weakened to IlUllw~.p(n ) + IlUllL~n) <<.C(K). m

PROOF. The proof is similar to that of Theorem 8.10. Each triple (/Zn, f n , U n ) (n E N) must satisfy also (fn, qgl) > 0 and either of the conditions (a) or (b) with ,L = )~l +/Zn and ~ = Xn E I2. Hence, fn ~ f in K as n --+ ~ combined with our assumption (f, qgl) ~ 0 imply (f, q91) > 0 as well. Next, instead of using Theorem 8.7, (8.27), one has to apply (the easier) Proposition 8.5, (8.18), to get

~n - - I t n l P - Z t n fs~ f ~ol dx + o(ItnlP-1).

Consequently,

I~n

1

Nonlinear spectral problems

459

for all n large enough, say, n ~> no. Recall that tnUn /> 1991 > 0 in 12 for all n ~> no, by Lemma 8.1. We conclude that 1 -Itn[-(P-2)lZn Un ~ ~ ( f , ~1)(/91 > 0

in S-2 for every n/> no.

But this fact violates both conditions (a) and (b) which require /Z n U()~n) ~ 0 for some ~,, E Z2. I-I Notice that for )~ - - ~,1 and p r 2, at least one of the conditions (a) or (b) in both, Theorem 8.10 and Theorem 8.11, is automatically satisfied. We state this result next as a simple consequence of a combination of Theorems 8.10 and 8.11 for the resonant problem (6.12), vis. --Apu--)~llulP-2u + f T(x)+f

"~pl(x)

inS2;

u--0

on 01-2,

(8.41)

where f T E L~(I-2) T and ( E R. It was shown originally in [57], Theorems 2.1 and 2.3, and [57], Theorems 2.6 and 2.8, for p > 2 and 1 < p < 2, respectively; see also [58], Theorems 3.2 and 3.6. We write f = f T + (q91 according to (5.1). COROLLARY 8.12. Let 1 < p < ~ , p ~ 2. Assume that K is a nonempty, weakly-star _L,L 2

compact set in L ~ ( 1 2 ) such that K n 79~ -- ~3 and (g, qgl) -- Of o r all g E K. Then we have: (i) There exists a constant C ( K ) > 0 such that, if f E K and if u E W l'p (12) is any weak solution to problem (8.41), then Ilullcl,~(~) ~< C ( K ) . (ii) Given a number 6 > O, there exists a constant C ( K , 3) > 0 such that, if f = f T + (q91 with f T E K and I(] >~ 6, then any weak solution to problem (8.41) satisfies

Ilullc,,~<~> ~< C ( K , 6). PROOF. Part (i) follows directly from Theorem 8.10. Part (ii) follows from Theorem 8.11, provided one allows only 6 ~< 1(1 ~< 6 t where 0 < 6 ~< 6 t < ~ are arbitrary, but fixed numbers. Hence, Iiullc~,~(~ ~< c ( g , ~, ~'). However, if we apply Corollary 8.8 instead of Theorem 8.11, we obtain part (ii) as it stands. [2 Theorem 8.11 has another important consequence, namely, the following improvement of the "classical" strong maximum and anti-maximum principles (cf. Theorem 4.4 and Remark 4.5) due to Arcoya and G~imez [5], Theorem 27, p. 1908, for K -- {f} r {0}. THEOREM 8.13. Let 1 < p < cxz. Assume that K is a nonempty, weakly-star compact set in L ~ ( S 2 ) that satisfies the following two conditions f o r each f E K: (i) (f, qgl) 5~ 0; and (ii) the resonantproblem (6.12) has no weak solution u E W~ 'p (12). Then there exists a constant 6 - 6(K), 0 < 6 < ~,2 - ~,1, such that f o r every f E K with fs2 f991 dx > O, any weak solution u E Wo 'p (12) to problem (1.7) satisfies the strong maximum and anti-maximum principles:

460

P. Tak6(

(SMP) u > 0 in ~ whenever ,kl - 6 < )~ < )~1; (AMP) u < 0 in S2 whenever )~l < )~ < ~1 § 6,

respectively. The corresponding result holds also if f ~ f qgl dx < O, with the reversed inequality ()~1 - )~)u < 0 in S2 wheneverO < 1)~- )~l[ < 6. PROOF. Denote

K+--{fEK:

fs2fg~

}.

In analogy with the proof of Theorem 8.10 above, let us fix a number 6 t with 0 < 6 t < )~2 -- ~.1, and write ,k = )~1 + #. Next, on the contrary to the conclusion of our theorem, suppose that there are three sequences {/Zn}nCX~=lC ( - ( ~ , 6t], {fn }L1 C K+, and {Un}7= 1 C

W~ 'p (~2), such that for each n = 1, 2 . . . . we have: (a) /Zn ~ 0 and/Zn ~ 0 as n ~ cx~; (b) Un is a weak solution of problem (1.7) with )~ = )~l + #n and f = fn; and (c) -lZn U(}n) <~0 for some Xn 6 S'2. The set K being weakly-star compact in L e~ ($2), we may extract a weakly-star convergent subsequence fn ~ f in L~(S-2) as n ~ cxz. Hence, f E K+. Now we apply Theorem 8.11(a) if/Zn < 0 or part (b) if/Zn > 0, to conclude that the sequence {Un}n~__l is bounded in C l ' ~ ( ~ ) . By Arzel?a-Ascoli's theorem in cl'r for any fixed /3t 6 (0, I3) this sequence contains a convergent subsequence Un --+ u in c l ' ~ ' ( ~ ) as n ~ cx~. Letting n ~ cx~ in the weak formulation of problem (1.7), with )~ = )~l +/Zn and with the pair (fn, Un) in place of (f, u), we arrive at (6.12) for the limit pair (f, u) = l i m n ~ ( f n , Un) obtained above. However, by our condition (ii), the resonant problem (6.12) has no weak solution. This contradiction finishes the proof of our theorem. D The nonexistence hypothesis for (6.12), i.e., condition (ii) in Theorem 8.13, is the topic of the last subsection (Section 8.6) in this section. We know from the Fredholm alternative at )~l, Theorem 6.7 (if p > 2) and Proposition 7.4 (if 1 < p < 2), that this nonexistence hypothesis fails for fs~ f~ol dx = 0. Moreover, Theorem 7.5(i) (if 1 < p < 2) requires only a weaker condition that guarantees nonexistence.

8.6. Nonexistence for )~ = ~1 Recalling Remark 4.3, here we slightly weaken the hypothesis 0 ~< f ~ 0 in S-2 in Theorem 4.2. The following result is due to Tak~i~ [57], Corollaries 2.4 and 2.9. THEOREM 8.14. Let 1 < p < o~. Givenanarbitraryfunction g ~ L~176

with O <<.g ~ 0 in ~2, there exists a constant y - y(g) > 0 with the following property: If f ~ L ~ ( ~ ) , f -~ 0, is such that f = fg. g +/g

with some

f g E ~ a n d / g E LCX~( ~ ) ,

461

Nonlinear spectral problems

and I l f g IIL~(S~)~ y [ f g l , then problem (6.12) has no weak solution u e Wo 'p (S-2).

Equivalently, given g as above, notice that there is an open cone C in L ~ (X2) with vertex at the origin (0 ~ C) such that g e C and problem (6.12) has no weak solution whenever fEC. PROOF OF THEOREM 8.1 4. On the contrary, suppose that there is a sequence of functions

{ f~ }~ 1 in L oc (S-2), fn ~ 0, such that fn -- f g ' g 1 Ilfgllt~
+ fg

with some

f g E IR and f g E L ~ ( n ) ,

and problem (6.12) with f -- fn has a weak solution u -

Un e

Wo 'p ( ~ ) , for each n -- 1, 2 . . . . . Taking advantage of the (p - 1)-homogeneity of problem (6.12), we may assume f ~ = 1 for all n ~> 1 without loss of generality. This means 1 that IIf g IIL~ (s2) ~< ~ and

in s

- A p U n - )~llunlP-Zun + g + f g

Un -- 0

on 0S2.

(8.42)

-g c~

We apply Corollary 8.12(ii), with K -- {g + fn }n=l ['J {g} and (g, q)l) > 0 to conclude that the sequence {Un}~__l must be bounded in C 1'/~( ~ ) . Fix any/3' e (0, 13) and invoke Arzel?~Ascoli's theorem in order to pass to a convergent subsequence Un --+ u in Cl'/~'(~). Thus, letting n --+ oo in the weak formulation of problem (8.42), we arrive at - A p U - )~1 lulp-2u + g ( x )

in I2;

u--0

on0S2.

But this equation has no weak solution by Theorem 4.2. We have obtained a contradiction and thus proved the theorem.

D

Theorem 8.14 has an important corollary which will be used later in Section 10. Taking g -- q)l we get: COROLLARY 8.15. Let 1 < p < co. Given an arbitrary function f ~- E L ~ ( I 2 ) with ( f - r , q)l) - O , there exist two numbers - o o < ( , < 0 < (* < cx~ with the following property: I f f -- f - r + (q)l with ( E IR \ [(,, (*], then problem (6.12) has no weak solution u e w~'P(s2). Notice that this corollary is in fact included in the proof of Corollary 8.8. It implies part (i) of Theorem 7.5 above.

9. A variational approach This section is concerned with a variational method introduced in [57], Section 7, and further explored in [59]. Our approach is motivated by the standard fact that, for any )~ < )~1

462

P. Tak6?

and any f e L ~ (S2), the functional J z defined in (7.1) is coercive and weakly lower

Wd'p

semicontinuous on (S-2). Since its coercivity is lost for ~ ) )~l, one naturally tries to resort to a minimax method. Often, a function u e L I(y2) will be decomposed as the orthogonal sum (6.1). Given a set .A//C L 1 (if2), we write

.A/l T de__f{tAT . tt = gll "(/91 + U T E.~/~ for some u II ~ IR and(u T 991)=0} In particular, if .A//is a linear subspace of L 1 (if2) with q91 E ./~, then we have

.A/~T -- {U E . ~ " (b/, qgl) - - 0 } .

9.1.

A minimaxmethod

We will show that a weak solution to problem (1.7) can be obtained by verifying that the "minimax" (or rather "maximin") expression

fl)~ de2 sup

inf

J z (rq)l -~- U T)

(9.1)

"CEI~ uT EwI'P (f2) T

provides a critical value flz for the energy functional J z ; cf. [51], Theorem 4.6, p. 24, for a related minimax method. More precisely, this will be the case if f = f T + ~.~Ol with f T e L ~ ( S 2 ) T and I~'l small enough depending on IIfTIIL~(S~), say Iffl ~< ~', either 1 < p < 2 and )~1 ~< )~ ~< ~.l + 8, or else 2 < p < cx~ and )~1 < )~ ~< )~l + 8. (Here, 8, 8 t > 0

and

W~'p

are sufficiently small constants.) Recall that (s T has been defined in (7.2). But we need to treat also the general case 1 < p < c~ and 1~ - )~l[ ~< 8 which leads us to investigate other expressions closely related to (9.1), such as the "localized" formulas

flZ de__f sup

inf

JZ (rqgl + uT),

(9.2)

J;~ (rgOl + UT),

(9.3)

a
def

/z~ --

inf

inf

a
where - ~ ~< a < b ~< c~ are suitably chosen numbers. All these variational formulas have one part in common, namely, the function

jz (r) de__f

min

Oqz(r~ol + u T)

of r 9 R.

(9.4)

uTEwI'P(~Q) T

Clearly, by basic arguments from the calculus of variations ([53], Theorems 1.2 and 1.3, p. 4), this function is well defined whenever )~ < A ~ , where the number A ~ has been introduced in (6.6), i.e., A ~ -- inf,

f [Vu[Pdx'u~W~'P(~)Twithf [u[Pdx-1].

(9.5)

Nonlinear spectral problems

463

Our method is based on the inequality Aoo > k l which we have shown in L e m m a 6.4. It justifies formula (9.1) for any fixed )~ < Aoo. We will see that the function j4" R --+ R is continuous (Lemma 9.1) with an easily computable asymptotic behavior as [rl --+ oo, depending on p, k and (. Consequently, its local minima and maxima, respectively, provide local minima and simple saddle points (hence, critical values) for the energy functional ,Yz, and thus also solutions to problem (1.7). Before starting formal arguments, we introduce the following notation. As both ,74 and j4 depend also on f ~ L ~ (f2), from now on we often write ,.74(u) ---- ,.74(u; f ) and j z ( r ) =j 4 ( r ; f ) to avoid possible confusion. Notice that, for u = rqgl + u T and f - (~pl + f T , with r, ( E R, u T E WJ'P(S2) T and f T E L~ q-, we have J4(u" , f ) =

,.7"4(r~ol + u q-", f q - ) - T(llgOlll 2L 2 ( S - 2 ) ,

j4 (r", f ) -- j4 (r", f T ) _ r (11qgl I12 L2(~2)

(9.6) (9.7)

9

It is now clear that it suffices to determine the properties of J4 and jz in the special case, ( = 0, i.e., for f - f T E L ~ ( I 2 ) T.

9.2. Asymptotic behavior o f the constrained minima In this paragraph we provide a simple sufficient condition for the criticality of flz (Lemma 9.2). This condition will be verified later in Section 9.3, L e m m a 9.7, for p ~ 2. We allow 1 < p < c~ and assume f E L ~ (~2) throughout the entire paragraph. Let 77 be an arbitrary, but fixed number with 0 < r / < A ~ - k l. We assume that 0 ~< )~ <~ A ~ - r/and f ~ L ~ ( 1 2 ) . Furthermore, in view of L e m m a 6.4 with 9 / = 77, we finda constant 0 < V~ < cxz large enough, so that A y~ >~ A ~ - 8977, and set

1(,

c - ~

Ay~

Notice that for any fixed r E R the functional u T w-~ J4 (rqgl + u -v) is coercive on the (closed linear) subspace W~ 'p ($2) T C Wo 'p (S2). This claim follows from the following inequalities which are valid whenever It] ~< T ~< 70-111u ~- []w~,p(~), for any fixed T E (0, cx~)" /.

/.

+ u )l dx

l[vu l[ Lp(s2) -- CT,

-

+ u )l

(9.8)

464

P Takdg

with another constant 0 < c r < oe depending solely on T. The first inequality in (9.8) is easily derived from formula (6.6). Consequently, any global minimizer u T r for the functional uTr-+ ffZ(r~01 + u T) on w0'P(s-2) T satisfies the estimate

Ilu~T Ilwl,~
~CT
where CT is a constant independent from )~ 9 [0, Aoo - 0] and r 9 [ - T , T]. Such a global minimizer always exists and verifies the Euler-Lagrange equations

T --Ap(z'qgl -t- U r ) -- XIZ'991 nt- uT( TIqP-2 )lr

-t- Ur)T

= f T ( x ) + ~r .~Ol(x) uT 77 = 0

(u vr ,

(9.9)

in S2, on 0S2,

qgl) -- 0,

with a Lagrange multiplier ~'r 9 R. Thus, the expression jz (r) in (9.4) is well defined. In our proofs of Theorems 9.8 and 9.11 in Sections 9.4 and 9.5, respectively, for 2 < p < oo we will show that the function jz :R -+ R attains a local minimum if s = s and a local minimum and two local maxima if )~l < )~ ~ )~1 -F 3, provided ~ > 0 is small enough. Similarly, for 1 < p < 2 (in the proofs of Theorems 9.8 and 9.10 in Sections 9.4 and 9.5), we will show that jz attains a global maximum if )~ = )~l, and a local maximum and two local minima if )~] - 6 ~< s < s with 6 > 0 small enough. Analogous arguments will be employed in the proofs of other following theorems as well. The following continuity properties of jz are proved in [57], Lemma 7.2, p. 222. LEMMA 9.1. Let 1 < p < cx~. The mapping (r,X, f ) w-> jx(r; f ) : N x [0, A o o - r/] x LCC(X2)w, -+ IR is continuous. In particular, if O < T < cx~ and K is a compact set in L~

{jx('; f)" [ - T , T] ~ R: (~., f ) 9 [0, Ace - r/] x K }

(9.10) then

(9.11)

is a family o f (uniformly) equicontinuous functions.

As usual, L ~ (12)w, stands for the Lebesgue space L ~ (g2) endowed with the weak-star topology. Obviously, if the function jz :R --+ R has a local minimum at some point r0 e R, and u~- is a global minimizer for the functional u T ~ Jz(r0~01 + u T) on w~'P(s2) T, then u0 -- r0q)l + u~- is a local minimizer for J z on Wo 'p (s and thus a weak solution to problem (1.7). Our next lemma displays a similar result if jz has a local maximum at r0 e R; it claims that fix in (9.2) is a critical value of J z . In spite of being a "maximin" method, it is related to Rabinowitz's "Saddle Point Theorem" [51], Theorem 4.6, p. 24, adapted to the functional J z . LEMMA 9.2 ([57], Lemma 7.4). Assume 1 < p < ec. Let 0 <. )~ <. A ~ - rl and let f e Lee(I-2) satisfy (f, ~ol) --O. Assume that the function jx :IR --+ IR attains a local maxi-

Nonlinear spectral problems

465

mum fl)~ at some point ro E IR. Then there exists u-~ E W~ 'p ([2) T such that u-~ is a global minimizer for the functional u T w-~ J)~(r0q)l + u T) on Wo'P(s2) T, u0 = r0q)l 4- u~ is a critical point for J z , and J z (uo) - fi)~. .!

REMARK 9.3. The function j z ' R --+ R is differentiable at ro with jz (to) = 0; see [57], Remark 7.5, p. 225. According to our following definition, u0 -- rotpl + u~- is a simple saddle point for J z . DEFINITION 9.4. A function uo E W~ 'p (~2) will be called a simple saddle point for J)~ (with respect to the orthogonal sum (6.1)) if uo - ro~01 + u~- is a critical point for J z , u~- is a global minimizer for the restricted energy functional u T w-~ J z (rOtPl + u T) on 1, p ([2)T, and the function jz "R ~ R attains a local maximum at to. This concept is tailored for our treatment of the energy functional J z defined in (7.1). A more general type of a saddle point is obtained in [51 ], Theorem 4.6, p. 24. REMARK 9.5. It is useful to notice that if f -- (~01 + f T with ( 6 IR and f T 6 LOC(s2)T, f T ~ 0 in [2, then jz(0) ~ jz~ (0) < 0 for ~,1 ~< )~ < A ~ . Indeed, it suffices to verify jz~ (0) < 0 for ( - 0, by (9.7). Let u~- be any global minimizer for the functional u T w+ ffzj (u T) on W; 'p ([2) T. Then j~ (0) ~< Jx ( u ( ) ~< Jx, (u~-) - j)~, (0)

for )~, <~ )~ < A ~ .

Making use of the Euler-Lagrange equations (9.9) with )~ ~,1, l" 0 and a Lagrange multiplier (o E R, we first compute the inner product (f, u~-) and then insert it into the right-hand side in (7.1) with )~ - )~l, thus arriving at =

(o) = -

- -

(.o -

( 1 - p1)(f

Ivu;

p dx

--

)~1

lu;

dx

)

< 0

(9.12)

unless u~- -- 0 in [2, by Lemma 6.4. However, u~- -- 0 in (9.9) would force also f T __ 0, a contradiction to our assumption f T ~ 0 in I2.

9.3. Asymptotic behavior of jz near 4-oo Given )~ < ~.1, it is quite natural to expect that the coercivity of the functional J z on

Wo'P(Y2) forces j z ( r ) ~

ec as Irl--+ oo. More precisely, we have the following result:

LEMMA 9.6. Let 1 < p < c~, 0 < ~ < ec, and let Hypothesis (HI) be satisfied. Assume that K is a nonempty, bounded set in L~ Then we have j z ( r ; f ) --+ cx) as Irl --+ oo uniformly for all ~. <. h l - 6 and f E K.

466

P. Tak6(

PROOF. Given r 6 R, )~ ~< ~.1 - - ~ and f E K, let u rT be any global minimizer for the functional uT ~--~Jz(rq)l + u T) on w~'P(s2)T; hence, j z ( r ; f ) -- ,.7z(ur; f ) ,

T

where U r - - "gq91 + U r .

-2 (Ur, qgl), w e have Owing to r = IlqglIIL2(I2)

Irl ~ Cllu~llwo,~(~2)

(9.13)

with a constant C > 0 independent from r E R. Furthermore, there is another constant C~ >~ 0 such that 1

,.7~(u; f)~>

z"0=~llullu~'~(x2)-q

for all u E Wl ' p (I2), k ~< )~1

-- ~

and f E K.

(9.14)

Finally, we combine (9.13) and (9.14) to conclude that j z ( r ; f ) = ,.7Z(Ur; f ) --+ cx)

as Irl ~ D

uniformly for )~ ~< )~l - ~ and f 6 K.

The linear degenerate boundary value problem (8.20) (i.e., (8.19)) plays a crucial role in our asymptotic formulas as Irl--+ cx~. Its solution set in 79~0~ has been described in Remark 8.6. The proofs of all our theorems in this section below rely heavily upon the following asymptotic results for JZl (r) as Irl--+ c~. Although we employ almost the same tools in both cases p > 2 and p < 2, primarily Corollary 8.8 and parts of its proof, the asymptotic behavior of jz~ near -+-cx~ is different in these two cases. Recall D qgl T -- {u E D~01" (U, qgl) m 0 } .

LEMMA 9.7. Let 1 < p < cx~, p ~ 2, and assume both Hypotheses (HI) and (H2). = 0 and

Let K be a nonempty, weakly-star compact set in L ~ (Y2) such that K n 79 • --991 (f, ~01) = 0 for all f E K. Then we have

[r[ p-2" jz, (r; f ) -+ - Q 0 ( V T, V T)

as

Irl ~

c~

(9.15)

uniformly for all f E K. Here, V T E D q91 T is the unique weak solution of problem (8.19) In particular, letting Irl--+ cx~ we have Jzl (r; f ) ~ 0 if p > 2 and Jzl (r; f ) --+ -cx~ if p < 2 uniformly for all f E K.

Notice that Proposition 5.4 guarantees Q0(V T, V T) > 0.

Nonlinear spectral problems

467

PROOF OF LEMMA 9.7. Suppose that (9.15) is false, i.e., there exist some sequences {rn}n~__l C R and {fn}n~=l C K with rn --+ +cxz as n --+ cx~, such that l i m ]lr,,Ip-2- J~-I (~'r/; f t / ) - { - Qo(Vr/--V, /7---+(7<)

vZ)l > o,

(9.16)

where Vnv E D (/91 T is the unique weak solution of problem (8 " 19) with f fn " Extracting suitable subsequences, we may assume rn --+ + ~ , the other case rn --+ -cxz being similar, and fn ~* f in L ~ (s as n --+ cx~. Consequently, the mapping f T ~ V T . {~pl}_k,L2 --+ 7)~01 being linear and compact, by Remark 8.6, we get also IlVnT - vTI[z~I ---> 0 as n --+ ~ where V T E D T is the unique weak solution of problem (8.19). In particular, ' q)l instead of (9.16) we may assume l i m (17p - 2 n--+ oo

jk,

(Z'n;fn)) =/k -~0(V T, vT).

(9.17)

Now recall formula (9.4); for each n -- 1, 2 . . . . . it yields T

Jkl ('t'n; fn) -- ~ l ('t'n(fll nt- Un ; fn)

T

l,p

with some u n ~ W o

( n ) T.

(9.18)

In addition, according to (9.9), u nT verifies

T u n v IP-2 (r~o, + u~)

=fn(x)+~n'q~l(X) T Un = 0

(9.19)

in S'2, on Og2,

(b/n-I-, (/91) = 0,

with a Lagrange multiplier (n 6 ~ . We apply Corollary 8.8 to get -2

nlimoo(rP-l~n) -- (p -- 2)ll~ol IlL2(n ) 9 Qo(V T V T) > 0.

T

1-p

Subsequently, writing vn -- rn

def

(9.20)

1 T

Vn - r~- u n , we may apply also Proposition 8.5 to con-

clude that a subsequence satisfies V, -+ V T strongly in D~0~ as n -+ oo. Next we write Vn - - (/91 "Jr- UnT and recall formula (5.9) to get T

"-" Tn2 ( p - l ) " A , _ ((491 + VnT ; l:n1 - P f n ) + ( f n , Wn)

=

2(p--l)

(q,

Y

) -

468

P Tak6g

which simplifies to

z'P-2" ~JX1('t'nqgl -+- tinv ; f.) + (f., V . ) - Q~z(V., V.).

(9.21)

Letting n --+ cx~ and applying Proposition 8.5 again, we conclude that

lim (rnp-2

g/---~oo

T in))_ Qo(VT, v T)

9 J z 1 (rn~Ol -~- u n ;

(/, v T)

=-Q0(v owing to (8.19). With regard to (9.18) we thus have lim (rnp - 2 . jx, (rn;

/1---+oo

fn))=-Qo(V T, VT)

which contradicts (9.17). The lemma is proved.

E]

Now we are ready to apply our variational method described in Section 9.1 in order to find critical points for the functional Jx given in (7.1). These critical points are precisely weak solutions to problem (1.7).

9.4. Existence of a solution f o r X near X1 Recalling the decomposition (5.1) we write f = f-r- + ~.qgl with f T K is as follows:

K and ~" 6 IR, where

HYPOTHESIS (H3). K is a nonempty, weakly-star compact set in L ~ ( I 2 ) , K C L~(~2) T, and such that 0 ~ K if p > 2 and K A D ~•0~ = 0 if 1 < p < 2. This hypothesis on K admits the possibility D / ~01 ' L 2 5~ {0} if 1 < p < 2. We begin with the following existence result which generalizes Theorem 6.7 if p > 2 and Theorem 7.5(ii), if 1 < p < 2. THEOREM 9.8. and (H3). Then whenever f T ~ (a) If p > 2

Let 1 < p < ~ , p ~ 2, and assume all three Hypotheses (HI), (H2) there exist positive constants 3 =- 8(K), 8' =- 8'(K) and C ( K ) such that, K and Iffl ~< ~', we have: and Z <<,X1 + 8, then the energy functional Jx possesses a local mini-

mizer u l ~ W; 'p (1-2) (hence, a weak solution to problem (1.7)) that satisfies Ilul IIc~,,(~) <~ C(K). (b) I f p < 2 and IX - Xll <<,8, then fix possesses a simple saddle point ul ~ w ~ ' P ( ~ ) that satisfies Ilul IIcl,~(~) ~< C ( K ) . Furthermore, if X <~ X1 - 8, the same bound holds f o r any global minimizer of fix.

Nonlinear spectralproblems

469

This theorem has been obtained in [59]: part (a) coincides with [59], Theorem 2.1 (for p > 2), and part (b) with [59], Theorem 2.8 (for p < 2). For all k < k l and ~" 6 II~, the conclusions of the theorem follow from the coercivity of the energy functional Jk, whereas for 0 < k - k l ~< 6 and any ~" 6 IR, it can be proved by a well-known argument employing topological degree; see [19], Theorem 14.18, p. 189. Finally, for k = kl it has been established in [34], Theorem 3.3, p. 958 (for p > 2), and [57], Theorems 2.2 and 2.6 (for any p ~ 2) if ~" = 0, and in [24], Theorem 1.1, p. 184 (for p < 2), and [58], Theorems 3.1 and 3.5 (for any p ~ 2) if Iffl ~< ~'. Based on our auxiliary results, above all Lemmas 9.1 and 9.7 and Remark 9.5, the proof of Theorem 9.8 is now only a matter of determining suitable (local or global) extrema of the function r ~ jk (r; f ) : I R --+ R defined in (9.4). PROOF OF THEOREM 9.8. Recall f = ~'r + f Y with f Y E K and I~l ~ ~'. For any fixed 6, 6' > 0, the conclusion of the theorem is evident provided k <~ k l -- 6, as a consequence of the coercivity of the functional ,.74 on W~ 'p ([2). By this argument, the bound Ilulllw~,p(s~) <~C(K, 6, 6') for a global minimizer u l of Jk is obtained first, with a constant C(K, 6, 6') > 0, and then the a priori regularity result from Proposition 2.1 is applied to the Euler equation (1.7) for Ul in place of u to get Ilul IIc,,~(~) ~< c ( g , ~, ~'), where C(K, 6, 6') > 0 is another constant independent from k and f , provided they satisfy k ~< kl -- 6, f T E K and Iffl ~< ~'. Details concerning regularity of Ul have been presented in Section 8.1. Now we look for constants 6, 6' > 0 sufficiently small in order to treat the case Ik - k ll ~< 6. We need to distinguish between the cases p > 2 and p < 2. Case p > 2. First, we combine L e m m a 9.1 with Remark 9.5 to conclude that M =-- M ( K ) de_f_ sup Jkl (0; f T ) > 0.

(9.22)

fTEK

Next we apply L e m m a 9.7 to get a constant T --= T (K) > 0 such that

I

1M

whenever Irl ~> T and f T 6 K.

(9.23)

Finally, we use these inequalities and invoke L e m m a 9.1 again to see that there exist constants 6, 6' > 0 such that 3 jk(0; f ) ~< - - M 4

1 < --M 2

~< j k ( + T ; f )

whenever Ik - kll ~< 6, f T 6 K and

with f = ffq91 -t- f T

I~1 ~ ~'.

(9.24)

Each function jk ('; f ) ' [ - T , T] --+ R being continuous, by L e m m a 9.1, it attains a local minimum at some point rz = rk ( f ) E ( - T , T). Recalling the definition of j k ( r ; f ) in (9.4), we conclude that the functional J k ( ' ; f ) on

WI'p (s

possesses a local minimizer Ul - rk .qgl + u T. We can employ the coer-

civity of the functional u T w-~ Jk('gq91 +

uT; f)

on wI'p(s

T uniformly for [rl ~< T,

470

P. Tak66

I~-~,ll ~ 8, f T ~ K and I~l ~ g; cf. inequality (9.8), in order to find a constant C ( K ) > 0 such that whenever I , k - )~ll <~ 8, f T ~ K and I~'1 ~< ~'. Conse-

IluTIIwd,p(s~)<~C(K)

quently, as above, well-known regularity results [3,18,45,62] (Proposition 2.1) guarantee Ilul IIc,:(~7) ~< C ( K ) as desired. Case p < 2. Let I)~ - ~.ll ~< 8, f T ~ K and Iffl ~ ~', with 6, 61 > 0 to be determined. From Lemmas 9.1 and 9.7 we infer the following facts: (i) /~(fT) dee = suprcR JXl (r; f T ) _ JXl ('r; f T ) for some r ----.~(fT) E R; (ii) /~K de2 i n f / v e K / ~ ( f T ) > --e~; (iii) there exists a constant T -- T (K) > 0 such that

jX1(~,; fT) ~ ~K -- 3 < /~K ~/i~(fT) = jx, (r; f T )

whenever Irl/> T and f T ~ K .

In particular, we get I?(fT)l < T for every f T e K. Combining Lemmas 9.1 and 9.7 once more, we can choose 8, 8' > 0 small enough to get jx(•

f ) ~< /~K -- 2 < /~K -- I ~< jx(r; f )

withf=~'gol+fT,

whenever 1 ~ - &l[ ~< 6, fm 6 K and 1~'[ ~< 6 I.

(9.25)

From these inequalities we deduce that each function jx ('; f ) " [ - T, T] ---> R attains a local maximum fix ( f ) at some point rx = rx ( f ) ~ ( - T , T). Clearly, we get also jx(-4-T; f ) +

1 <~ f i x ( f ) = jx(rx; f ) .

(9.26)

Now, the existence of a simple saddle point u 1 E W01'p (s for the functional ,,74 follows from Lemma 9.2 with ul = rxqgl + u T. The bound Ilul IIc,,~(~) ~< C ( K ) is obtained in the same way as for p > 2. The proof is complete. O

9.5. Existence of two or three solutions Our second theorem is a multiplicity result for the resonant value )~ = )~1. Although it is taken from [58], Theorems 3.1 and 3.5, its present form is more specific in the qualitative description of solutions. Closely related results have been obtained in [21 ], Theorems 1.1 and 1.2, by different arguments. THEOREM 9.9. Let 1 < p < o0, p 7/= 2, and assume all three Hypotheses (H1), (H2) and (H3). Then there exists a constant 81 =_ 6f(K) > 0 such that problem (6.12) with f _~ f T + ~ 991 has at least two (distinct) weak solutions specified as follows, whenever f T E K and 0 < I~1 ~< ~': The energy functional fix1 (which is unbounded from below) possesses a local minimizer and a simple saddle point.

Nonlinear spectral problems

471

PROOF. We simply continue our reasoning from the proof of Theorem 9.8 with ~. = ~1. This time we focus our attention on the function jz~ (r; f ) for [r[ ~> T and 0 < [~'] ~< 6'. Recalling formula (9.7) we observe that it suffices to treat the case 0 < ~" ~< 61; the other case - 6 f ~< ~" < 0 is completely analogous. Thus, let us consider arbitrary 0 < ~" ~< 6 f and

fTEK. Case p > 2. We employ Lemma 9.7 to see that there is a number T I = T I(K) ~> T such that j)~ (r" f-r) ~< 1 for all r ~> T I. Hence, using (9.7) and setting M f de__f(M + 1).

-2 [[~1 ][LZ(.Q) we have

j~.l (l.., f ) = j)~l (.g; f-l-) _ r~.ll~o1112 L2(g2) ~< - M

forall r ~> Tr ae=fmax{TI, Mf/r

(9.27)

Gathering all inequalities from (9.24) and (9.27) we arrive at

J~-I (0; f ) -

1

J)~l (0; f T ) ~< - M < - - M 2 1

J~-I (Tr f ) ~< - M < - - M 2

~< j)~, (-l-T" f ) ' '

~< jz, (T" f ) .

(9.28) (9.29)

Since JZl('; f ) : R - - + IK is continuous, by Lemma 9.1, it attains a local minimum #Zl ~< - M at some point rz 1 _= rz~ ( f ) E ( - T , T) and a local maximum/3z~ ~> - 8 9 at another point r[1 z, ( f ) 6 (0, T~). We obtain a local minimizer ul - r Z l

WI'p (s

9 ~01 + u~- for the functional Jz~(.; f ) on

exactly as in the proof of Theorem 9.8(a), with )~ -- )~1. Finally, the existence

of a simple saddle point u2 E wI'p(s

for Jz~ follows from Lemma 9.2 with u 2 -

l

"g)~l "q)l -t- U2T.

Case p < 2. Recall that 0 < ( ~< 6 f and f T E K are arbitrary. We employ Lemma 9.7 (formula (9.15)) to see that there is a number T f ----TI(K) >~T such that j)~, (r; f T ) ~> --2. Q0(w, w ) - I r l 2-p

for all Irl > Z'.

Recall that the numbers ~ = r (f-r), fix and T = T (K) have been defined in the proof of Theorem 9.8(b). Hence, using (9.7) and setting

-2 M' def 2./~/( Ilqgl IIL2(~) and

-2 M " def 4. Qo(w, w)IlqOl IILZ(n),

472

P. Tak6(

we have j),, (r; f ) -- J)vl (Z'; f T ) _ r~"

I1~0~1122(n)

) - 2 . Qo(w, w ) . Irl 2.p - r e II~01112 L2(,Q) 1 ~> - ~ r~" Ilq)l l122(n)/>/~K

(9.30)

for all r ~< - T o where T~ defmax{T', M ' / ~ , ( M tt/~ ) l / (p-1)}. Gathering all inequalities from (9.25) and (9.30) we arrive at

J~.l (4-T;

f ) ~/~K -- 2
J)vl (-T~-; f ) } .

(9.31)

Since jz~ (.; f ) : R ~ R is continuous, by Lemma 9.1, it attains a local maximum flZl t> /~K -- 1 at some point rZl -------rZl ( f ) 6 ( - T , T) and a local minimum #Zl ~
U]

The following two theorems on the existence of at least three solutions to the Dirichlet problem (1.7) have been established by Tak~i6 [59]. First, we consider the subcritical case ~.l - ~ ~< )~ < )~l. The following theorem corresponds to [59], Theorems 2.3 and 2.10, for p > 2 and p < 2, respectively. THEOREM 9.10. Let all three Hypotheses (H1), (H2) and (H3) be satisfied. If 2 < p < cx~, there exists a constant 6' = 8'(K) > 0 such that, for any d E (0, ~'), there is another constant 6 -~ 6(K, d) > 0 such thatproblem (1.7) with f -~ f T + ~ol has at least three (pairwise distinct) weak solutions specified as follows, whenever )~1 - ~ < )~1, f T E K and d <, ]~'l ~< 6': The functional J z (which is bounded from below) possesses two (distinct) local minimizers of which at least one is global, and a simple saddle point. If 1 < p < 2, there exist constants ~ -~ 8(K) > 0 and 8' =- 8'(K) > 0 such that problem (1.7) with f -~ f T + ~ qgl has at least three (pairwise distinct) weak solutions specified as above, whenever )~1 - ~ <. )~ < ~l, f T ~ K and 1~'1 ~< 8'. PROOF. Case p > 2. We further continue our reasoning from the proofs of Theorems 9.8 and 9.9. Let d E (0, 8') be arbitrary, but fixed. Without loss of generality, we may assume d <~ ~" ~< 6' which in turn implies T <<. T' <<. Tr <<. Td. We still need to choose 8, 8' > 0 small enough; of course, both independent from d. From Lemma 9.1 and the first part of (9.29) we deduce that jz(Td; f ) <~ - 3 M whenever I~. - )~11 ~< 8, f T ~ K and d <~ Ir ~< W, provided 8, 8' > 0 are sufficiently small. On the other hand, given any

Nonlinear spectral problems

473

)~ ~ [)~1 - 8, ~.l), L e m m a 9.6 guarantees that there is a number TJ 4) -- TJ z) (K, 8, 8') > Td such that j z ( r ; f ) ~> 1 for all equalities, we arrive at

Irl/> T~z~,

f - r E K and d ~< I~'1 ~< ~'. Collecting these in-

3 jz (Td; f ) ~< - - M < 1 ~< j4 (_+_~,.(z) ld ; f ) 4 whenever )~1 - 8 ~< ~ < Jkl, f - r E K and d ~< Iffl ~

~'.

(9.32)

As in the proof of Theorem 9.9, since jz ('; f ) " R --+ R is continuous ( L e m m a 9.1) and satisfies inequalities (9.24) and (9.32), it attains a local minimum #~ ~< -43-M at some point r)~ = r4 ( f ) E ( - T, T), a local m a x i m u m f14 ~> - 89M at another point r~ _= r~ ( f ) (0, Td), and another local minimum/24 ~< -43-M at some point fz ---- r z ( f ) 6 (T, Td(Z)). We obtain a local minimizer U l - r4 9 q)! + u~- and a simple saddle point u t1 r~ 9q)l + (u'l) T for the functional J x ( - ; f ) on w;'P(Y2) exactly as in the proof of Theorem 9.9. Another local minimizer u2 6 Wo 'p (S2) for J z is obtained with u2 - f'4 "q)l + u~-. The conclusion of the theorem follows by setting u3 = u t1" Case p < 2. We continue the procedure commenced in the proof of Theorem 9.8. Given any ~ 6 [)~1 - 8, )~1), from L e m m a 9.6 we deduce that there is a number T (~) = TOO(K, 6') > T such that j 4 ( r ; f ) ~> fiK for all Irl/> T (x), f T 6 K and I~'1 ~< ~'. Combining this fact with inequalities (9.25) we get j4(-t-T; f ) <~ fiK -- 2 < fiK -- 1 ~< min{j4(?; f ) , j 4 ( + T ( 4 ) ; f ) }

whenever ~.1 - 8 ~< ~. < )~l, f - r ~ K and

I~1 ~< ~'.

(9.33)

According to the proof of Theorem 9.8, each function j)~(.; f ) " [ - T ()0, T OO] ~ R attains a local m a x i m u m fix ( f ) at some point r)~ ---- r)~( f ) 6 ( - T, T) and moreover, by (9.33), also t local minima #4, #4t t ~
T

Finally, we treat the supercritical case )~l < )~ ~< ~l + 8. The following theorem is taken from [59], Theorems 2.4 and 2.11, for p > 2 and p < 2, respectively. THEOREM 9.1 1. Let all three Hypotheses (HI), (H2) and (H3) be satisfied. I f 2 < p < ec, there exist constants 8 - 8 ( K ) > 0 and 8 t - 8 t ( K ) > 0 such that problem (1.7) with f - f q- + ~ q)l has at least three (pairwise distinct) weak solutions specified as follows, whenever )~l < )~ <~ )~1 + 8, f T ~ K and ]~'[ <~ 8 t" The functional ffz (which is unbounded from below) possesses a local minimizer and two (distinct) simple saddle points. l f l < p < 2, there exists a constant 8 t - St(K) > 0 such that, f o r any d ~ (0, 8'), there is another constant 8 -- 8(K, d) > 0 such thatproblem (1.7) with f =- f ~- + ~q)l has at least

474

P. Takd(

three (pairwise distinct) weak solutions specified as above, whenever )~1 < ~. ~ ~.1 -t- 6, f-r- ~ K and d <, I~'1 ~< a'. PROOF. Case p > 2. We return to the end of the proof of Theorem 9.8 with ~.1 < )~ ~< ~.l + 6, f 7 ~ K and Iffl ~< a'. We keep the same constants 6, 6' > 0 also for the rest of the current proof. From (9.4) and (9.7) we obtain j4(r; f ) =

r~" II~01112L2(n)~

j4(r; f T ) _ = plrl p

IVq)l

dx-)~

J 4 (Z'qgl ; f - r ) -

q)~dx

2 ~< - - - -1( ) , -- ~.1)I z" Ip -+- 6' Irl" Ilgolllt2(s2),

rff II~~ II2L2(s~)

- rffll~01ll2 r E R.

(9.34)

P

fs21Vq)llPdx =

Recall f s 2 q ~ f d x - 1 and T(4)(K, 6') > T such that j4(r; f ) ~ - M

for all

)~l. Hence, there is a number T (4) ----

Irl >/T (4), f T E K and ICI ~< g.

(9.35)

The constants M -- M ( K ) > 0 and T ---- T ( K ) > 0 have been determined by (9.22) and (9.23), respectively. According to the proof of Theorem 9.8, the function j4('; f ) : R ~ R attains a local minimum #4 ~< - 8 8 at some point r4 -- r4 ( f ) 6 ( - T , T), by (9.24). Furthermore, combining (9.24) and (9.35) we conclude that j4('; f ) attains also local maxima fl~, fl~' ~> - 89M at two points r~ ------r~ ( f ) and r Z -- r~' ( f ) , respectively, where - T OO < r~ < 0 < r~: < T (4) . It follows that the functional ,74('; f ) on Wo'P(I-2) possesses a local minimizer Ul -- r4 .~01 + u qBy Lemma 9.2, J4 ('" f ) has also two (distinct) simple saddle points 1 " f

u2 = r 4 "~Ol + u~- and u3 = r~' 9qgl + u~-. Case p < 2. We follow a pattern of steps similar to the proof of Theorem 9.10. We further continue our arguments from the proofs of Theorems 9.8 and 9.10. Let d 6 (0, 6') be arbitrary, but fixed. Again, we may assume d ~< ~" ~< 6' which entails T ~< T' ~< T~ ~< Td. The constants 6, 6' > 0 will be chosen small enough as follows, both independent from d. 5 From L e m m a 9.1 and the second part of (9.31) we deduce that j4 (--T d; f ) >~ /~/( 4 whenever 1)~ - )~11 ~< 6, fT- 6 K and d <~ Iffl ~< a', provided 6, 6' > 0 are sufficiently small. On the other hand, given any )~ 6 ()~l,)~l + 6], (9.34) guarantees that there is a number Td(4) -- Td(4) (K, 6, 6') > Td such that j4(r; f ) ~ Zd(Z), f T 6 K and d ~< Iffl <~ a'. We collect these inequalities to get

j4(4-T(4); f) <~ flK

5

-- 2 <~fl K - -~ <~ j4 (-- Td ; f)

whenever ~.l <~. ~<,kl + 6 , f T 6 K a n d d ~< Iffl ~*.

(9.36)

Recalling the proof of Theorem 9.10, since j4 ('; f ) " IR --+ IR is continuous (Lemma 9.1) and satisfies inequalities (9.25) and (9.36), it attains a local maximum fi4 ~>/~/~ - 1 at

Nonlinearspectralproblems

475

some point rz ---- rz ( f ) E ( - T , T), a local m i n i m u m / z z ~< fix - 2 at another point r~ ---r~ ( f ) E ( - T d , ~?), and another local maximum/~z ~> fiK - 5 at some point fz -- f)~ ( f ) E ( - T ~ z) - T ) We obtain a simple saddle point Ul -

rz 9 ~ol + u T and a local minimizer

w;'P(I2) exactly as in the proof of Theorem 9.9. Another simple saddle point u2 E Wo'P(s2) for 3"z is obtained with u'1 = r [ . (pj + (U'l) T for the functional J z ( ' ; f ) on

U 2 - - "~L 9 qgl +

U2T. The proof is concluded by setting u3 - u'1.

if]

10. (Un)ordered pairs of sub-/supersolutions In this section we apply monotonicity methods to investigate the solvability of the resonant problem (6.12)with f = f - r + ~'q)l, for a fixed function f T E L ~ ( I 2 ) w i t h (f, q)l) = 0 a n d for any ~" E R. Among other results, we will prove Theorem 7.5 in both cases, 1 < p < 2 and 2 < p < ec. More precisely, we take advantage of the weak comparison principle for the Dirichlet p-Laplacian (Lemma 2.4) and apply it to the Dirichlet problem (8.41) with an arbitrary parameter ~" ~ R. We use the fact that, for any x E I2 fixed, the function f(x) = f T ( x ) + ~. q)l(x) is strictly increasing with respect to the parameter ~" E R, by q)l (x) > 0. Consequently, a weak solution u E W; 'p (S2) to problem (8.41) becomes a strict subsolution (or supersolution, respectively) to (8.41) if ~" is increased (or decreased). In this way we will obtain an unordered family of solutions for ~'# < ~" < ~"#, in addition to those solutions for ~', ~< ~" ~< ~'* described in Theorem 7.5. Here, ~', ~< ~'# < 0 < ~'# ~< ~'* are suitable numbers which are described below. The method of strict sub- and supersolutions for this kind of elliptic problems was developed by De Coster and Henrard [15], Section 8. The following standard definition is used in [15], Section 8. DEFINITION 10.1. A function u inequalities,

E

C 1( ~ ) is called a

subsolution of problem (6.12) if the

{ fs2 IVul p-2vu "Vvdx <<.Xl fs2 lul p-2uvdx + fs? f (x)vdx, u~<0

(10.1)

on 0s

hold for all v E Wo 'p (s with v ~> 0 a.e. in s A supersolution ofproblem (6.12) is defined analogously with both inequalities in (10.1) reversed. For our monotonicity method it will turn out to be convenient to work with the inverse of - A p. Given any f E L oc (s the energy functional

Jo(u) de--f l fs? lVulP dx P

J's? f (x)u dx

u E Wo 'p (S2),

476

P Tatcaa

is strictly convex and coercive on Wo 'p (S2). Therefore, 3"o has a unique (global) minimizer u f which is also the unique critical point of J0, i.e., the corresponding Euler problem

- A p u -- f ( x )

in S2;

u--0

on aS2,

(10.2)

possesses a unique weak solution in Wo'P(s2) equal to uf. We have u f E C1'/3(~) for some/3 E (0, or), by regularity (Proposition 2.1). Thus, we may define the nonlinear mapping ( - - A p ) - I 9L~~ --+ L~176 by putting

(_Ap)_l f def u f. This mapping is continuous and takes bounded sets from L ~ (S2) to bounded sets in C I ' ~ ( ~ ) . This implies that ( - A p ) -1 is a completely continuous selfmapping of Le~(S-2), i.e., it maps bounded sets to relatively compact sets. Furthermore, the standard weak comparison principle (Lemma 2.4) shows that ( - A p ) -1 is orderpreserving (or monotone), that is, for all f, g e L ~ ( I 2 ) , f ~< g in 12 implies (--Ap) -1 f ~< ( - A p ) - l g . Finaly, ( - A p ) -1 being an inverse, it is even strictly order-preserving (or strictly monotone), that is, f ~< g and f _~ g in L~ imply ( - - A p ) - l f ~< ( - - A p ) - l g and ( - - A p ) - l f ~k (--Ap)-lg. Next, given any fixed function f T E L ~176 (12)T and a parameter ~ E IR, we define the fixed point mapping Tr 9L ~ ( ~ ) --+ L~162 by -

-

Tfu def = ( - A p ) -I(x 1lulP-2u 4- f-r + ~'q91)

for u E L ~ (12).

(10.3)

Clearly, TCu = u if and only if u is a weak solution of problem (8.41). The mapping TC is strictly order-preserving, one-to-one, continuous and takes bounded sets from L ~ (S2) to bounded sets in C I , ~ ( ~ ) . Moreover, for every u E L~(~2), ~'l < ~'2 in R implies T~u <<, T~2u and T~I u ~ TO2u in S-2. REMARK 10.2. The following weaker notions of sub- and supersolutions to problem (6.12) are in fact sufficient for our purposes: Let f = f T + ~ol E L~(S-2), where ( f 7 , qgl) = 0 and ~" E R. A function u E L~(~2) is called a subsolution of problem (6.12) if Tr u >~ u in 12. Similarly, u is called a supersolution if Tr u ~< u in S2. Finally, to shorten our notation, given r E R, let us denote by F ( r ) the set of all pairs (~', u T) 6 R • Wo 'p (S2) T satisfying the boundary value problem (8.41) with u -- r~01 + u T. Notice that F ( r ) 5~ 0, because the Euler-Lagrange equations (9.9) with )~ = ~l have a solutionu T r E W01,p (12)T, with a suitable number (r E ~. The asymptotic behavior of F ( r ) as r --+ icx~ has been determined in Proposition 8.5 and Corollary 8.8. Throughout this section we assume 1 < p < c~, p r 2, the hypotheses (HI) for any p _L,L2 and (H2) for p > 2, together with f T ~ 0 in I2 for p > 2 and f T ~ D~0~ for p < 2.

10.1. Existence results using ordered pairs The aim of this paragraph is to construct a family of solutions in Theorem 7.5 (to problem (8.41)) in the following two cases: (i) 2 < p < cx~ and ~'. ~< ~" ~< ~'*; and (ii) 1 < p < 2

477

Nonlinear spectral problems

and (# ~< ( ~< ( # with ( g= 0. Well-ordered families of solutions to problem (8.41) can be obtained using the fixed point theory for strictly order-preserving mappings in L ~176 (S2); see, e.g., [ 14], [40], Chapter I, or [56]. We begin with an auxiliary existence result:

LEMMA 10.3. Assume thatO ~ (o E IR and uo E L~176 satisfy Tr o = uo. Let ( E ~ be such that either 0 < ( < (o or else (o < ( < O, and let also 0 < r < oo. Then there exist (i, ti E IR and ui E L~176 (i = 1, 2) with the followingproperties: (i) (1 < 0 < (2 if p > 2 ((2 < 0 < (1 if p < 2, respectively) with I(il < l( I, and t! < 0 < t2 with I t / l < r , f o r i = 1, 2; (ii) Tciui -- ui -- tZ 1(q)l + l)7) E C l'fi ( ~ ) with 1)7 ~ -1991 f o r i -- 1,2; (iii) u l ~ uo ~ u2 in S-2. In particular, the following inequalities hold throughout I-2: Ul ~ T(ul ~ T(uo ~ uo ~ u2 ~ T(u2

if 0 < ( < (o,

(10.4)

T(ul ~ Ul ~ uo ~ T(uo ~ T(u2 ~ u2

ifr < ~ < o.

(lO.5)

PROOF. This lemma is an easy consequence of Corollary 8.8 and its proof. Indeed, take ro > 0 small enough, ro <~ r, such that first, for any pair ((i, u [ ) E F(t/-1) (i -- 1, 2), 0 < Itil < ro implies ICil < ICI, by (8.35), and second, uVi - t [ - l v 7 satisfies tiui -- qgl -+v~-/> 89q)l, by the proof of Corollary 8.8. Recall that F (t/-1) ~ 0. The sign of the right-hand side of (8.35) determines (1 < 0 < (2 if p > 2 and (2 < 0 < (1 if p < 2. Third, take r~ > 0

even smaller, r~ ~< to, such that r~luol ~< 89 in S-2. It follows that - r ~ < tl < 0 < t2 < r~ implies Ul ~< uo ~< ue in S-2. Finally, from T~iui = ui (i = 0, 1, 2) and part (i) we deduce (10.4) and (10.5). [-1 We can now apply monotone iterations to construct well-ordered families of solutions. PROPOSITION 10.4. Assume that ai E R and gi E L~(S'2) (i = 1,2) satisfy a2 ~ 0 <, al if p > 2 (a2 < 0 < a l if p < 2, respectively) and Tai Zi = Zi. Let 0 < r < oo. Then we have the following three statements: (a) There exist (1, tl E R such that (1 < 0 if p > 2 (0 < (j < al if p < 2), - r < tl < 0, Tr

+v~-)ECI't~(,Q)

and

Ul <~Zl

inS2.

Moreover, f o r any ( ~ [(1,all, the sequence Ul <~ T~ul ~ ... ~ T~ul ~ ... (<. Zl) converges in L~ to some Wl(() with T~wl(() = Wl(() and Ul <. Wl(() ~< Zl in S-2. The function Wl : [ ( 1 , a l l --+ L~ is strictly monotone and continuous from the left. For each (0 E [(1, al), the right-hand limit Wl ( ( ) xa wl ((0+) exists in L~ as ( x a (0 and satisfies T~o Wl ((o+) - Wl ((o+) and Wl ((0) ~< Wl ((o+) in S-2. (b) There exist (2, t2 E IR such that (2 > 0 if p > 2 (a2 < (2 < 0 if p < 2), 0 < t2 < r,

Zf2u2 -- u2 -- t2 1 (qgl + 1)~-) E C l ' f l ( ~ )

and

z2 <. u2

in ~2.

Moreover, f o r any ( E [a2, ~2], the sequence u2 >/T(u2 >/ ... >/T?u2 >/ ... (>/z2) converges in L~176 to some w2(() with T~w2(() -- w2(() and Z2 ~ W2(~') ~ U2 in $2. The

478

P. Tak6(

function to2 : [a2, if2] ~ L~(~2) is strictly monotone and continuous from the right. For each ~o ~ (a2, ~'2], the left-hand limit wz(ff) ,,7 w2(~'o-) exists in LC~(S-2) as ~ /7 ~o and satisfies T~ow2(~'o-) = w2(~'o-) and w2(~'o-) ~< w2(~'o) in S2. (c) The numbers ~i, ti ~ ~ (i -- 1, 2) in parts (a) and (b) can be chosen such that Ul <~ zi <~ u2 in ~ (i = 1,2) and, if p > 2, then also Wl(~') ~< w2(~') for all ~ [~'1, al] A [a2, ~'2]. In general, it might happen that Wl(~'o) ~ Wl(ffo+), tO2(ff0) ~ tO2(ff0--) or Wl(ff) w2(~') in S2. Notice that 0 6 (~'1, al] N [a2, ~'2) if p > 2. PROOF OF PROPOSITION 10.4. We commence with the proof of part (a). The existence of

~'1, tl 6 • with the desired properties follows from Lemma 10.3. Given any ~" 6 [~'1, al], we have u 1 =Tr u 1 ~< TCu I ~ T~- Z 1 ~ Tal Z 1 - - Z1 in I-2, by the monotonicity properties of the mapping (~, u) ~ Tr Since Tr is a completely continuous self-mapping of L ~ ( I 2 ) , the convergence of the bounded, monotone increasing sequence u I <~ Tr u 1 ~< "" ~< T~nu 1 ~< 9.. (~< Zl) to some wl(~') in L ~ ( I 2 ) follows. Clearly, we have TCwl(~') = Wl(~') and Ul ~< Wl (~') ~< Zl in 12. Using the monotonicity properties of (~', u) w-~ Tr u again, for any ~'1 ~< ~" < ~'" ~< al we obtain Wl (~") ~< Wl (~'") and wl (~") ~ 1131(~tt) in 12. Now take any ~'o ~ (~'1, all. The left-hand limit Wl(~')/z Wl(~'0-) exists in L~(S2) as ~-/z ~'o and satisfies Tr Wl (~'o-) = Wl (if0-) and Ul ~< Wl (~'o-) ~< Wl (~'o) in ~2. Here, we have employed the fact that the inverse ( - A p ) - 1 is a completely continuous, strictly orderpreserving self-mapping of L ~ ( I 2 ) . On the other hand, our definition T~oUl / z Wl(~'o) in L~(S'2) as n --+ cx~ implies Wl(~'o) ~< Wl(~'o-) in ~2. These inequalities show that Wl (~'0-) = Wl (~'o) in I2 as claimed. Similarly, given any ~'o 6 [~'1, al), the right-hand limit Wl(~') ~ Wl(~'o+) exists in L~(S2) as ~" "~ ~'o and satisfies T~owl(~o+) = Wl(~'o+) and Wl (~'o) ~< Wl (~'o+) in S2. Part (b) is proved analogously as part (a) by reversing the inequalities for ~"'s and u's. Part (c) is verified by choosing [ti[ small enough in parts (a) and (b), so that u 1 ~< zi <<,u2 in S2 (i = 1, 2). If p > 2 then, for ~" ~ [~'1, al] N [a2, ~'2] and every integer n ~> 1, we obtain Ul <<,T~ul <<,T~u2 <<,u2 in ~2. Finally, letting n --+ cx~ we arrive at Wl (~') ~< w2(~') in ~2. Fq

10.2. Existence results using unordered pairs In contrast with the preceding paragraph, here we construct a family of solutions in Theorem 7.5 (to problem (8.41)) in the following two remaining cases: (iii) 2 < p < cxz and ~# <~ ~ ~< if# with ~" ~ 0; and (iv) ~', ~< ~" ~< ~'*. Recall that ~', ~< ~'# < 0 < ~# ~< ~* are some numbers depending on f T . This family of solutions is unordered and is obtained by the method of strict sub- and supersolutions developed in [ 15], Section 8. We recall that the right-hand side of (8.41) is a strictly monotone increasing mapping in ff 6 I~. The following existence result, based on unordered sub- and supersolutions, is a special case of [15], Theorem 8.2, p. 448; cf. [24], Lemma 2.4, p. 191:

Nonlinear spectral problems

479

LEMMA 10.5. Let u_ and fi be sub- and supersolutions of problem (6.12), respectively, such that u(xo) > u(xo) f o r some xo E ,(2. Then problem (6.12)possesses at least one weak

solution u E Wo'P(s2) in the closure (relative to the norm of Cl (~2)) of the set

S de~---f{U E w I ' P ( s - 2 ) ( ] C I ( O )

9

U(X1) < /~(Xl), U(X2) > /~(X2)for s o m e Xl, x2 E if2 }. Notice that here, in contrast to (10.4) and (10.5), i.e.,

ul <<. Tcul <<. Tcuo <~ uo

if 0 < ~" < ~'0;

uo ~ T(uo ~ T~u2 ~ u2

i f f 0 < ~" < 0 ,

respectively, the sub- and supersolutions must not satisfy u ~< fi in Y2. From L e m m a 10.5 we can easily derive the following" COROLLARY 10.6. Let ~l < ~ < ~2 be real numbers. Assume that Ui E C 1( ~ ) (i = 1, 2) satisfy T~iui = ui and Ul (xo) > u2(xo) f o r some xo E Y2. Then the fixed point problem T~u -- u possesses at least one solution u in the closure (relative to the norm of C 1(E))) of the set

s

{u Wo

c'

U(X1) < Ul (X1), U(X2) > u z ( x z ) f o r s o m e Xl, X2 E if2}. PROOF. Notice that Ul -- Tr <<.T~ul and T~u2 ~ T~2u2 = u2 in ~ . Consequently, we may apply L e m m a 10.5 to get a solution to the equation T~ u -- u in the closure of the set ,S. D It is now obvious that, letting ~" range over the entire interval [~'l, ~2] in Corollary 10.6 above, we obtain a family of functions u r in the closure of the set ,S such that Tr u ~ -- u for each ~'. Of course, we set ur = ui. Due to the hypothesis u~l (x0) > u~2 (x0) for some x0 E Y2, we cannot have u~, ~< u~,, in ~ for all ~" and ~'" satisfying ~1 ~ ~.f ~ ~,, ~ ~'2.

10.3. (Un)ordered sets o f solutions f o r ~. = ~l N o w we are ready to prove a generalized version of T h e o r e m 7.5 in all four cases (i)-(iv) described above. THEOREM 10.7. Let 1 < p < r and f T q~ D_L,L2

p :fi 2. A s s u m e t h a t f T E L~(S-2) satisfies ( f T , qgl) - - 0

991 9 (i) Then there exist two constants ~',, ~* E R depending on f T , ~., < 0 < ~'*, such

thatproblem (8.41)possesses at least one weak solution u E Wo'P (s2) if and only if ~, <~

~ <<.~*.

P. Tak6(

480

(ii) Moreover, there are two additional constants ~#, ~# 6 R depending on f q- again, ~, <~ ~# < 0 < ~# <~ ~*, such thatproblem (8.41)possesses at least two distinctweak solutions in W I'p (I2) provided ~# < ~ < ~# and ~ # O. (iii) Let 1 < p < 2. Given any g~- ~ C o ( ~ ) with (g7, ~Pl) - 0 and g7 ~ 0 in S2, there exists a number p =_ p ( g 7 ) > 0 such thatproblem (6.12) has at least one solution whenever f ~ L~ satisfies IIf - gTIIL~(n) < P. Parts (i) and (iii) of this theorem, for 1 < p < 2, are essentially due to Dr~ibek and Holubovfi [24], T h e o r e m 1.1, p. 184. Parts (i) and (ii), for any p # 2, where f T is fixed, are due to Takfi6 [58], Theorems 3.1 and 3.5. However, in the case of one space dimension ( N = 1), i.e., when S'2 = (0, a) is a bounded interval in R l, even a stronger result with ~', = ~'# < 0 < ~'# = ~'* was shown earlier in [22], T h e o r e m 1.3. They succeeded to show that the two types of solutions, obtained by applying Proposition 10.4 and Corollary 10.6, respectively, are indeed distinct whenever ~, < ~ < ~* and ~ # 0. For N ~> 2 we are unable to verify if they are distinct also for ~, < ~ ~< ~# or ~ # ~< ~ < ~*. PROOF OF THEOREM 10.7. We c o m m e n c e with the proof of part (i), namely, with the existence of a solution to problem (8.41) for ~', ~< ~" ~< ~'*. We first invoke F ( r ) 7~ 0 for r 6 R. Hence, problem (8.41) has a solution u = r~01 + u T, where ~" 6 ]I{ is a suitable number. This means that T~ u = u. From now on we distinguish between the cases 2 < p < cxz and 1 < p < 2 . Case p > 2. According to formula (8.35) in Corollary 8.8, we can find a number r > 0 such that sign ~" = signt whenever 0 < Itl < v. In particular, the hypotheses of Proposition 10.4 are verified with some numbers a2 < 0 < a l. Let us consider ~'* - -

sup Z

and

~',

-- infZ,

where Z -- {( E R" Tr u -- u for some u ~ L ~ ($2)].

(10.6)

Clearly, these numbers must be finite by formula (8.35) combined with (8.41) (or by Corollary 8.12). Consequently, we have -cx~ < ~', ~< a2 < 0 < al ~< ~'* < c~. Notice that Tr = u* and Tr = u, for some u*, u, 6 L~(I-2), by Corollary 8.12 and continuity. Next, let us take al = ~'* and a2 = ~', in Proposition 10.4. Making use of parts (a) and (b) of this proposition, where

u

c*] -

c*],

we have completed the proof of part (i) of our theorem for p > 2. In addition, both functions Wl : [~'1, ~'*] ~ L ~ ( s and w2 : [~',, ~'2] --+ L ~ (I-2) obtained there are strictly monotone, bounded, and satisfy Tr Wl (~') = Wl (~') for ~'l ~ ~" ~< ~'* and T~ w2 (~') = w2(~') for ~', ~< ~" ~< ~'2. Hence, there is a constant 0 < M < co independent from ~" and i = 1, 2 such that Ilwi(~)llfl,y (~) <~ M, by regularity. To prove part (ii) of T h e o r e m 10.7, we will employ Corollary 10.6. First, let us fix a n u m b e r (; with (, < (; < 0 and - ( ; small enough, so that U'l (t'l)-1 ((/91 + (V'l) T) is a -

-

Nonlinear spectralproblems

481

solution of problem (8.41) with ~'~ in place of ~'. By L e m m a 10.3, this can be achieved by ! choosing - t 1' small enough, 0 < -t~ < r, such that , 1 1 u I ~< x ( t l ) - qgl ~< 2wi(ff) z-

for all ff a n d / -

1,2.

(10.7)

!

f

!

Repeating this procedure, we fix another number ~'~ with ~'1 < ~'2 < 0 and -~'2 small enough, so that u'2 - (t~) -1 (qgl + (v2) , 7-) is a solution of problem (8.41) with ~'~ in place of ~'. Again, - t 2' is chosen to be small enough, t 1' < t 2' < 0, such that u 2' < u '1 inl-2 Notice that TC/ Ui' - - U'i for i -- 1 2. So we may apply Corollary 10.6 to conclude that, given ' ~'~), the fixed point problem TCu -- u possesses at least one solution u in the any ~" 6 (~'1, closure (relative to the norm of C 1( ~ ) ) of the set

U(Xl)

<

Uq(Xl),U(X2)

! > Uz(X2 )

for some Xl,X2 E S-2}.

Inequalities (10.7) guarantee l/)i ( i f ) ~ S for all ~" and i -- 1, 2. Finally, taking ~'# -- ~'~ and letting ~'~/7 0, we obtain part (ii) of the theorem. The number ~'# > 0 is obtained in a similar way. The proof of the theorem for p > 2 is now complete. Case p < 2. Again, according to formula (8.35), we can find a number r > 0 such that sign ~" = - s i g n t whenever 0 < Itl < r. The hypotheses of Proposition 10.4 are verified with some numbers a2 < 0 < as. The numbers ~'* and ~', defined in (10.6) are finite by the same reasoning as for p > 2. So again -cx~ < ~', ~< a2 < 0 < a l ~< ~'* < cx~ together with Tr = u* and T ~ , u , -- u , for some u*, u , E L ~ (s In contrast with the proof for p > 2 above, here we may have to interchange the roles of well-ordered and unordered families of solutions. If u , ~< u* in s then a strictly m o n o t o n e function w : [~',, ~'*] L ~ ( S 2 ) is constructed by m o n o t o n e iterations in the same way as in the proof of part (a) or (b) of Proposition 10.4, such that T~ w(~') = w(~') and u, ~< w(~') ~< u* in s for every ~" 6 [~',, ~'*]. On the other hand, if u , ( x o ) > u*(xo) for some x0 6 S-2, then we deduce from Corollary 10.6 that, given any ~" E (~',, ~'*), the fixed point problem T~u = u has at least one solution u in the closure (relative to the norm of C 1( ~ ) ) of the set

U(Xl) < u , ( x l ) , u(x2) > u*(x2) for some Xl,X2 ES2}. This proves part (i) of our theorem for p < 2. N o w we prove part (ii). First, let us fix a sufficiently small number al with 0 < al < ~'*, so that Zl - (tl)-m (qgl + (v'l) T) is a solution of problem (8.41) with as in place of ~'. By ! L e m m a 10.3, this can be achieved by choosing - t 1' small enough, 0 < - t 1 < r, such that

1 -1 U* Zl ~< ~(t',) ~01 ~ < 2 . m i n l u . , }

inl-2.

Making use of part (a) of Proposition 10.4, where 0 problem (8.41) has a weak solution Wl (() E Wo 'p (s

(10.8) <

(1 < as < (*, we conclude that

whenever ~1 ~< ( ~< as. This solution

P Tak6(

482

satisfies wl (~') ~< z l in E2, and consequently, inequalities (10.8) guarantee that it is different from the one obtained in part (i). Indeed, if u, ~< u* in E2, then the strictly monotone function w 9[~',, ~'*] --+ L ~ (S2) from the proof of part (i) satisfies wl (~') ~< zl < u, ~< w(~') in S2 for every ~" ~ [~',, ~'*]. On the other hand, if u,(xo) > u*(xo) for some x0 6 S-2, then inequalities (10.8) entail wl (~') ~ S for ~'1 ~< ~" ~< al. Again, part (ii) of the theorem is obtained by taking ~'# = al and letting ~'1 "~ 0. The number ~'# > 0 is obtained similarly. To prove part (iii), let 1 < p < 2. Fix gT 6 C O( ~ ) with (gT, ~Ol) = 0 and gT ~ 0 in $2. Given a number r/6 ~, define the function

go(x)

de f gT (x) + r/,

x E 1"2.

If r/is such that 0 < I~1 ~/9, where p > 0 is taken sufficiently small, then problem (6.12) with f g+0 has at least one weak solution, by Proposition 7.4. Now consider any f 6 L~(E2) with IIf - gTIIL~S2~ ~< /9 or, equivalently, with g_p <~ f <~ g,o in ~ . Define the family of functions ho ~ L~(E2) for - 1 ~< 0 ~< 1 by

ho def { f + 0 ( f -- g_p) =

f+O(gp--f)

if-l~<0~<0, if0~<0~
Notice that h+l = g+p and h0 -- f , and ho is strictly monotone increasing in 0 E [ - 1 , 1]. We can apply the same fixed point method as in the proof of the existence in part (i) above, with the mapping T~ replaced by To" L ~ (S2) --+ L ~ (E2),

Toll~'def_ ( - - A p ) -l()~,lulP-2u+ho)l

for u 6 L ~ ($2).

(10.9)

Clearly, Tou = u if and only if u is a weak solution of problem (6.12) with ho in place of f . Similarly as in part (i), such a solution exists whenever - 1 ~< 0 ~< 1. In particular, 0 = 0 yields the desired result. We have finished the proof of the theorem. 73

11. Bifurcations and the Fredholm alternative

In this section we keep the setting from the previous one" We assume 1 < p < c~, p ~ 2, Hypotheses (HI) for any p and (H2) for p > 2, together with f T ~ 0 in E2 for p > 2 and f T r 79• for p < 2. Recall from the beginning ofthe previous section that F ( r ) denotes --q91 the set of all pairs (if, u T) ~ R x Wo 'p ($2) T verifying problem (8.41) with u = rq91 + uT; we have F (r) :~ 0 for every r ~ R. If )~ -- )~l and p ~ 2, we have seen that the asymptotic behavior of"large" solutions u - r(qgl + v x) to problem (8.41) with a parameter ff E R, as r --+ + c ~ , is described by Proposition 8.5 and formula (8.35). In addition, Theorem 10.7 asserts that at least one solution exists precisely when if, ~< ~" ~< ~'*, and another (different) solution exists when ~'# < ~" < if# and ~" ~ 0, where -cx~ < ~', ~< ~'# < 0 < ~'# ~< if* < cx~ are some constants depending on f T . Therefore, a natural question to ask is if the set

.~- U {'r, F(r)} C R x R x w~'P(~)T 'rER

Nonlinearspectralproblems

483

contains a connected subset C such that (rn, ~'n, u~) 6 C for some (~'n, UnT ) 6 F (rn) and for some sequences rn --+ 4-oo. In other words, we wish to investigate global bifurcations of large solutions u -- r(qvl + v -r) from :tzoc (Deimling [16], Section 28.5, p. 387). Since this question apparently requires the Lyapunov-Schmidt reduction method to be applied to problem (8.7), with/Zn -- 0 and tn E R, it isout of the scope of the present article. Instead, we will regard X = X1 + / z (# 6 IR) as the bifurcation parameter and give a solution to a somewhat easier global bifurcation problem; see Dmibek [19], Chapter 5, and Dmibek et al. [23], Section 5. Last but not least, let us mention that if the parameter X stays strictly below the second eigenvalue )~2, say X - )~1 -[" /Z ~ )~2 -- 6 for some 6 > 0, then owing to Lemma 8.1, any bifurcation branch of large solutions to problem

--ApU = X[ulP-2u -q- f-l-(x) + ~" .qgl (x)

in S-2;

u-0

on0S-2,

(11.1)

contains only strictly positive or strictly negative solutions when they bifurcate from infinity. Near infinity, their sign can be determined by various methods which are based mostly on the results from Sections 8.5 and 8.6 (see [23], Section 5). Here, either X or ( can play the role of the bifurcation parameter, while the other one is held fixed.

11.1.

An abstract global bifurcation result 1

Under a solution of (1.7) we now understand a pair (X,u) of X E IR and u 6 w0'P(I2) satisfying the integral equality

fs2 [Vu[p-2(Vu " Vr

- X fs2 [u[P-ZuCdx-- /'s2 f Cdx

(11.2)

for every r ~ W1' p (S-2). Let X -- Wo'P(I2) and let X ' - - W - I ' P ' ( S 2 ) s t a n d for its dual space with the duality pairing (., .) between X and X' induced by the inner product from L2(S2). Then (11.2) is equivalent to the abstract operator equation

Z(u) - XS(u) -- F,

(11.3)

where Z, S" X ~ X' and F 6 X' are defined as follows, for any u, r 6 X"

(Z(U), r

fs2 [VulP-2(Vu" V~)dx,

(~(u), r

]~ [u[p-2ur dx and

(F, r

f r dx.

It is proved in [19], Chapter 5, that the operator Z - XS satisfies condition or(X) from Skrypnik [52] (which is nothing else but condition (S+) from Browder and Petryshyn [7])

484

P. Takt(

and thus, its (topological) degree can be defined. Namely, if Br (0) denotes the open ball in X centered at the origin with radius r > 0, then the degree of the mapping u w-~ 2-(u) AS(u) - F on Br(O), denoted by Deg[2- - AS - F; Br (0),

0],

is well defined provided the equation 2-(u) - AS(u) = F has no solution u 9 OBr(O). Letus note that the basic properties of the degree "Deg" are the same as those of the well-known Leray-Schauder degree "deg" for mappings from X into itself [52]. Here, 2-(.) replaces the identity mapping on X, whereas AS(.) + F plays the role of a compact perturbation. DEFINITION 1 1.1. Let/zo 9 II~. We say that (/zo, c~) is an asymptotic bifurcation point of (11.3) if there exists a sequence of pairs {(/Zn, Un)}n~__l C R x X such that (11.3) holds with (A, u) -- (#n, un) for all n - 1, 2 . . . . . and 0Zn, IlUn IIx) ~ 0Z0, ~ ) as n -+ ~ . For u 9 X, u ~ 0, set v - Ilullx2u. Then problem (11.3) is equivalent to ~(v) - AS(v) - ]Iv]]2(p-1)F, and so the term

~(A, V) des { 0]]V]]2x(P-1)F

if v r O, if v - O ,

which is in fact independent from A 9 ]R, represents a compact perturbation "of higher order" ( = 2(p - 1)) in the variable v in the equation 27(v) - ,~S(v) - g(,~, v).

(11.4)

It follows immediately from this transformation that the pair (#0, ~ ) is an asymptotic bifurcation point for (11.3) if and only if (#0, 0) is a bifurcation point (from the set of trivial solutions) for (11.4). For C c II~ • X we define (the set) C to be the closure in IR x X of the set of all pairs (#, v) 9 IR x X such that v r 0 and (#, IIv llx 2 v) ~ C. In particular, applying Lemma 8.1 to problem (11.2), we may speak about two asymptotic bifurcation points (A 1, -bcx~). In [19], Theorem 14.18, it was proved that (A1,0) is a bifurcation point for (11.4). Let us reformulate this result in terms of our problem (11.3). PROPOSITION 1 1.2. Let F 9 X t, F ~ O. Then the pair (Al, oo) is an asymptotic bifurcation point f o r (11.3). Moreover, there exists a maximal closed set C c R x X (in the ordering by set inclusion), such that C is connected in R x X, and C has the following properties: (i) there exists a sequence {(/d,n,/,/n)}n~_l C C such that (/d,n, IlUnllX) - * (A1, oo); (ii) either C is unbounded in the X-direction, or else there exists an eigenvalue Xo of-Ap such that Xo > A1 and there is a sequence {(l~n, Un)}n~=l C C satisfying 0Zn, IlUn IIx) -+ 0~0, ~ ) .

Nonlinear spectral problems

485

REMARK 1 1.3. The assumption F --/=0 (which corresponds to f ~ 0 in (1.7)) implies that problem (11.3) cannot have the trivial solution u = 0. Consequently, C contains no sequence of pairs (#k, uk) with (#~, Ilukllx) ~ @, 0). Hence, the statement of Proposition 11.2 follows directly from [19], Theorem 14.18, using the transformation u w-~ v = Ilullx2u.

11.2. Bifurcations from infinity We begin with a result from Dmibek et al. [23], Theorem 5.2, about the behavior of two branches of solutions to problem (11.2) bifurcating from -+-cx~and their possible intersection. THEOREM 1 1.4. Let F E X', F 7~ O. Then there is a pair o f maximal closed sets C +, C - c It{ x X o f solutions to (11.3) such that both sets C + and C - are connected in IR x X, where C+ denote the closures in IR x X o f the respective sets o f all pairs (lZ, v) E IR x X satisfying v ~ 0 and (lZ, v/llvll 2) E C -~, and moreover, C + have the following properties: (a) there exist sequences o f pairs (lZn, Un) E C + and (lZ~n, U'n) E C - such that ~ A1 ,

/Zn

lZ !

n

~ 1.1,

Ilunllx --~ m

and

! Ilunlls --+ ec,

together with !

Un

- -

Ilun IIx

-~

qgl

1199111x

and

Un

Ilu,,'llx

--+

(t91

Ilqgl IIx

(11.5)

strongly in X as n --+ oc; (b) either both C + and C - are unbounded in the X-direction, or else C+ Cl C - contains a point other than {(X l, 0) }.

Some remarks are in order. REMARK 1 1.5. (i) We can combine Lemma 8.1 with Theorem I 1.4 to conclude that the convergence in (11.5) is strong in C I , ~ ' ( ~ ) as n ~ oc. (ii) Part (a) clearly specifies what we mean under the asymptotic bifurcation points 0~, +ec). (iii) Part (b) is valuable if one can show that at least one of the bifurcation branches C + is bounded in the A-direction and C+ A C- contains no point {(A0, 0)} such that A0 is an eigenvalue of - A p with X0 > A1. This implies C + N C- ~ 0 and therefore C + U C- is a connected set which connects -cx~ with + e c in the sense described in part (a). PROOF OF THEOREM 11.4. Upon the transformation V n - un/llunll 2, the statement of the theorem follows directly from Corollary 8.12 (a priori bounds) and [19], Theorem 14.20. The limits in (11.5) follow immediately from Lemma 8.1. [3

486

P. T a k 6 6

Next, we investigate the bifurcation branches C + c N x X obtained in Theorem 11.4. The a priori bounds established in Theorems 8.10 and 8.11 allow us to detect whether ). belongs to the left or the right neighborhood of ).! provided ()., u) 6 C + and the norm Ilullcl,~(~7) (equivalently, Ilullg~(s~)) is large enough. For such u we write u = t -1 x (q)l nt- V-J-) with 0 7~ t ~ IR and

fs2 vr

('D1 d x - - O.

COROLLARY 1 1.6. Let 1 < p < oo, p ~ 2, and let 0 < 8

< ).2 -

).1

and 0 <<.~ < 00.

Assume that f T E L ~ ( Y 2 ) is given with fs2 f T~ol dx - - 0 and f q- ~ D q•) l

"

Moreover,

let C + be as in Theorem ll.4. Then there exists a constant M E IR, C ( { f T } ) ~< M < ( C ( { f T}) being the constantfrom Theorem 8.10), such that f o r e v e r y p a i r ()., u) ~ C + U C with -~. ~< ). ~< ).2 -- 8, ]lullc1,/~(~) > M , and u - - t - 1 (q)l -4- I)T), we have (p - 2) x (). ).l) > 0 and t u > 0 in X'2. The same conclusion remains true if the hypothesis ). >~ -~. is dropped and the condition

Ilullf~,e(~) > M is strengthened to Ilullwl,p(s?) + llulIL~(S2) > M. This corollary is due to [23], Corollary 5.8. PROOF O F C O R O L L A R Y 11.6. We prove only the case p > 2; the proof for p < 2 is similar. So let p > 2. By Remark 11.5(i), there is a constant M > 0 (sufficiently large) with the following property: If ()., u) ~ C + U C - with - ~ ~< ). ~< ).2 - 8, Ilullc,,e(~) > M, and u = t -1 (q)l nt- vY), then we have also tu > 0 in S2. Taking M ~> C ( { f T } ) large enough, we must have ). > ). 1 by Theorem 8.10(a). [3 We finish with the following result established in [23], Corollary 5.6, as well. It complements Corollary 11.6. COROLLARY 1 1.7. Let 0 < 8 < ),2 given with fs2 f gol dx 7~ O. Moreover, a constant M E N, C ( { f } ) <~ M < oo such that f o r every pair ()., u) E C + U

).1 and 0 <<.~ < cx~. Assume that f ~ Lee(Y2) is let C + be as in Theorem 11.4. Then there exists ( C ( { f } ) being the constant from Theorem 8.11), C - with - ~ ~< ). ~< ).2 - 8, [lU[Ic~.~(~) > M , and

u = t -1 (~01 + v~-), we have: (i) (a) ( f , gol) > O, ()., u) ~ C +, and t (b) ( f , ~ol ) > O, ()., u) ~ C - , and t (ii) (a) ( f , ~ol) < O, ()., u) ~ C +, and t (b) ( f , ~Ol) < O, ()., u) ~ C - , and t The hypothesis ). >~ - ) . may be dropped

> 0 =~ u > 0 in Y2 and ). < ).l ; < 0 =:~ u < 0 in s and ). > ). l. > 0 =~ u > 0 in Y2 and ). > ).1; < 0 =~ u < 0 in Y2 and ). < ).l. if the condition Ilu ]]c~,~(U) > M is strengthened

to Ilullwd,~(s2) + llullL~(re) > M.

PROOF. We prove (i)(a) only, the other cases being similar. By Remark 11.5(i), there is a constant M > 0 (sufficiently large) with the following property: If (X, u) 6 C + with -~. ~< ;~ <~ z2 - ,~, Ilullc,,e(~) > M, and t > 0 in u - t-l(qgl + VT), then we have also u > 0 in S-2. Taking M ~> C ( { f } ) large enough, we must have X < ).l owing to Theorem 8.11, condition (b). V]

Nonlinear spectral problems

487

A number of other similar results for the bifurcation branches C + ~ C~, n o w depending on the parameter ~" ~ • in f = f T + ~-991, can be found in [23], Section 5.3, together with the corresponding figures.

Acknowledgments This work was supported in part by the German Academic Exchange Service (DAAD, Germany) within the exchange programs "PROCOPE" and "Acciones Integradas" with France and Spain, and by the Federal Ministry for Education and Research (BMBF, Germany) through its International Office, Grant No. CZE-01/004.

References [ 1] W. Allegretto and Y.-X. Huang, A Picone's identity for the p-Laplacian and applications, Nonlinear Anal. 32(7) (1998), 819-830. [2] A. Anane, Simplicit6 et isolation de la premikre valeur propre du p-laplacien avec poids, C. R. Acad. Sci. Paris, Strie 1305 (1987), 725-728. [3] A. Anane, Etude des valeurs propres et de la r~sonance pour l'op~rateur p-Laplacien, Th~se de doctorat, Universit6 Libre de Bruxelles, Brussels (1988). [4] A. Anane and N. Tsouli, On the second eigenvalue of the p-Laplacian, Nonlinear Partial Differential Equations, Proc. Conf. Fbs, Morocco, May 9-14, 1994, A. Benkirane and J.-E Gossez, eds, Pitman Res. Notes Math., Vol. 343, Longman, Essex, U.K. (1996), 1-9. [5] D. Arcoya and J.L. G~rnez, Bifurcation theory and related problems: Anti-maximum principle and resonance, Comm. Partial Differential Equations 26 (2001), 1879-1911. [6] I. Birindelli, Hopf's lemma and anti-maximum principle in general domains, J. Differential Equations 119 (1995), 450-472. [7] EE. Browder and W.V. Petryshyn, Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces, J. Funct. Anal. 3 (1969), 217-245. [8] B.M. Brown and W. Reichel, Computing eigenvalues and Fu({k-spectrum of the radially symmetric p-Laplacian, J. Comput. Appl. Math. 148(1) (2002), 183-211. [9] Ph. C16ment and L.A. Peletier, An anti-maximum principle for second order elliptic operators, J. Differential Equations 34 (1979), 218-229. [ 10] Ph. C16ment, J. Fleckinger, R.F. Mamisevich and F. de Thtlin, Existence ofpositive solutions for a nonvariational quasilinear elliptic system, J. Differential Equations 166 (2000), 455-477. [ 11 ] M. Cuesta, D.G. De Figueiredo and J.-P. Gossez, The beginning of the Fu(ik spectrum for the p-Laplacian, J. Differential Equations 159 (1999), 212-238. [12] M. Cuesta and E Tak(t~, A strong comparison principle for the Dirichlet p-Laplacian, Proc. Conf. ReactionDiffusion Equations, 1995, Trieste, Italy, G. Caristi and E. Mitidieri, eds, Lecture Notes in Pure and Appl. Math., Vol. 194, Dekker, New York (1998), 79-87. [13] M. Cuesta and E Tak~, A strong comparison principle for positive solutions of degenerate elliptic equations, Differential Integral Equations 13 (4-6) (2000), 721-746. [14] E.N. Dancer and E Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine Angew. Math. 419 (1991), 125-139. [15] C. De Coster and M. Henrard, Existence and localization of solution for second order elliptic B VP in presence of lower and upper solutions without any order, J. Differential Equations 145 (1998), 420-452. [ 16] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin-Heidelberg-New York (1985). [17] J.I. Dfaz and J.E. Saa, Existence et unicit~ de solutions positives pour certaines ~quations elliptiques quasilin~aires, C. R. Acad. Sci. Paris, Strie 13t)5 (1987), 521-524.

488

P Takdg

[18] E. DiBenedetto, C 1+~ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7(8) (1983), 827-850. [19] E Drfibek, Solvability and Bifurcations of Nonlinear Equations, Pitman Res. Notes Math., Vol. 264, Longman, Essex, U.K. (1992). [20] E Drfibek, Nonlinear eigenvalue problems and Fredholm alternative, Nonlinear Differential Equations, P. Drfibek, E Krej~i and E Takfi~, eds, Chapman & Hall/CRC Res. Notes Math., Vol. 404, CRC Press LLC, Boca Raton, FL (1999), 1-46. [21] P. Drfibek, Geometry of the energy functional and the Fredholm alternative for the p-Laplacian in higher dimensions, Electron. J. Differ. Equ. Conf. 08 (2002), 103-120. [22] E Drfibek, E Girg and R.E Manfisevich, Generic Fredholm alternative-type results for the one-dimensional p-Laplacian, Nonlinear Differ. Equ. Appl. 8 (2001), 285-298. [23] E Drfibek, P. Girg, P. Takfi~ and M. Ulm, The Fredholm alternative for the p-Laplacian: Bifurcation from infinity, existence and multiplicity of solutions, Indiana Univ. Math. J. 53 (2) (2004), to appear. [24] E Drfibek and G. Holubovfi, Fredholm alternative for the p-Laplacian in higher dimensions, J. Math. Anal. Appl. 263 (2001), 182-194. [25] E Dr~bek, P. Krej~f and P. Takfi~, eds, Nonlinear Differential Equations, Chapman & Hall/CRC Res. Notes Math., Vol. 404, CRC Press LLC, Boca Raton, FL (1999). (Formerly Pitman Mathematics Series.) [26] E Drfibek and S.B. Robinson, Resonance problems for the p-Laplacian, J. Funct. Anal. 169 (1999), 189-200. [27] J. Fleckinger, J.-E Gossez, E Takfi6 and E de Th61in, Existence, nonexistence et principe de l'antimaximum pour le p-Laplacien, C. R. Acad. Sci. Paris, S6rie 1321 (1995), 731-734. [28] J. Fleckinger, J.-P. Gossez, E Takfi~ and E de Th61in, Nonexistence of solutions and an anti-maximum principle for cooperative systems with the p-Laplacian, Math. Nachr. 194 (1998), 49-78. [29] J. Fleckinger, J. Hern~dez, P. Takfi6 and E de Th61in, Uniqueness and positivity for solutions of equations with the p-Laplacian, Proc. Conf. Reaction-Diffusion Equations, 1995, Trieste, Italy, G. Caristi and E. Mitidieri, eds, Lecture Notes in Pure and App. Math., Vol. 194, Marcel Dekker, New York-Basel (1998), 141-155. [30] J. Fleckinger, J. Hern~qdez and E de Th61in, Principe du maximum pour un systkme elliptique non lin~aire, C. R. Acad. Sci. Paris, S6rie 1 314 (1992), 665-668. [31 ] J. Fleckinger, J. Hern~ndez and E de Th61in, On maximum principles and existence ofpositive solutions for some cooperative elliptic systems, Differential Integral Equations 8 (1995), 69-85. [32] J. Fleckinger and P. Takfi6, Unicit~ de la solution d'un syst~me non lin~aire strictement coop~ratif, C. R. Acad. Sci. Paris, S6rie 1 319 (1994), 447-450. [33] J. Fleckinger and P. TakS.~, Uniqueness of positive solutions for nonlinear cooperative systems with the p-Laplacian, Indiana Univ. Math. J. 43 (4) (1994), 1227-1253. [34] J. Fleckinger and E Takfi6, An improved Poincar~ inequality and the p-Laplacian at resonance for p > 2, Adv. Differential Equations 7(8) (2002), 951-971. [35] S. Fu6~, Solvability of Nonlinear Equations and Boundary Value Problems, Reidel, Dordrecht, The Netherlands (1980). [36] S. F u ~ , M. Ku6era and J. Ne6as, Ranges of nonlinear asymptotically linear operators, J. Differential Equations 17 (1975), 375-394. [37] S. Fu6ik, J. Ne6as, J. Sou~ek and V. Sou6ek, Spectral Analysis of Nonlinear Operators, Lecture Notes in Math., Vol. 346, Springer-Verlag, New York-Berlin-Heidelberg (1973). [38] J.L. G~rnez and E Girg, Fredholm Alternative for the p-Laplacian at the first eigenvalue and bifurcations from the infinity, Preprint, University of West Bohemia (2000). [39] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York-Berlin-Heidelberg (1977). [40] E Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math., Vol. 247, Longman, Essex, U.K. (1991). [41] T. Idogawa and M. Otani, The first eigenvalues of some abstract elliptic operators, Funkcial. Ekvac. 38 (1995), 1-9. [42] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss., Vol. 132, Springer-Verlag, New York-Berlin-Heidelberg (1980). [43] A. Kufner, Weighted Sobolev Spaces, Teubner-Texte Math., Vol. 31, Teubner, Stuttgart (1980).

Nonlinear spectral problems

489

[44] A. Kufner, A.O. John and S. Fu61"k,Function Spaces, Academia, Prague (1977). [45] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12(11) (1988), 1203-1219. [46] E Lindqvist, On the equation div(IVu]p-2Vu) -k- )~lu]p-2u : 0, Proc. Amer. Math. Soc. 109(1) (1990), 157-164. [47] R.E Man~isevich and P. Tak~i6, On the Fredholm alternative for the p-Laplacian in one dimension, Proc. London Math. Soc. 84 (2002), 324-342. [48] M.A. del Pino, E Dr~ibek and R.E Man~isevich, The Fredholm alternative at the first eigenvalue for the one-dimensional p-Laplacian, J. Differential Equations 151 (1999), 386-419. [49] M.A. del Pino, M. Elgueta, and R.E Man~isevich, A homotopic deformation along p of a Leray-Schauder degree result and existence for (]uflP-Zut) t -k- f (t, u) = O, u(O) = u(T) = 0, p > 1, J. Differential Equations 80(1) (1989), 1-13. [50] S.I. Pohozaev, The fibering method in nonlinear variational problems, Topological and Variational Methods for Nonlinear Boundary Value Problems, P. Dr~ibek, ed., Pitman Res. Notes Math., Vol. 365, Addison Wesley Longman, Essex, U.K. (1997), 35-88. [51] EH. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. Vol. 65, Amer. Math. Soc., Providence, RI (1986). [52] I.V. Skrypnik, Nonlinear Elliptic Boundary Value Problems, Naukovaja Durnka, Kyiev (1973) (in Russian). English transl.: Teubner-Texte Math., Vol. 91 (1986). [53] M. Struwe, Variational Methods, 2nd Edition, Springer-Verlag, New York-Berlin-Heidelberg (1996). [54] G. Sweers, Hopf's lemma and two-dimensional domains with corners, Rend. Istit. Mat. Univ. Trieste 28 (Suppl.) (1996), 383-419. [55] E Takfi6, Convergence in the part metric for discrete dynamical systems in ordered topological cones, Nonlinear Anal. 26(11) (1996), 1753-1777. [56] E Tak~6, Degenerate elliptic equations in ordered Banach spaces and applications, Nonlinear Differential Equations, E Drfibek, E Krej~f and E Tak~i~, eds, Chapman & Hall/CRC Res. Notes Math., Vol. 404, CRC Press LLC, Boca Raton, FL (1999), 111-196. [57] E TaMi~, On the Fredholm alternative for the p-Laplacian at the first eigenvalue, Indiana Univ. Math. J. 51(1) (2002), 187-237. [58] E Take6, On the number and structure of solutions for a Fredholm alternative with the p-Laplacian, J. Differential Equations 185 (2002), 306-347. [59] E Takfl6, A variational approach to the Fredholm alternative for the p-Laplacian near the first eigenvalue, Preprint (2002). [60] E Tak~i6, L. Tello and M. Ulm, Variational problems with a p-homogeneous energy, Positivity 6(1) (2001), 75-94. [61 ] E Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations 8(7) (1983), 773-817. [62] E Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126-150. [63] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, New YorkAmsterdam-Oxford (1978). [64] G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York (1987). [65] J.L. V~izquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202. [66] K. Yosida, Functional Analysis, 6th Edition, Grundlehren Math. Wiss., Vol. 123, Springer-Verlag, New York-Berlin-Heidelberg (1980).