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A reduction method for some nonlinear dirichlet problems
0362-546X/79
Non/mm Analysis, Theory, Merhods & Appkofmns, Vol. 3, No 6, pp. 795-813 0 Pergamon Press Ltd. 1979 Prmted m Great Bntam
A REDUCTION
METHOD
FOR SOME NONLINEAR PROBLEMS
1101-0795 SOZ.oO/O
DIRICHLET
KLAUS THEWS Mathematisches
Seminar
der Christian
Albrechts-Universitlt, (Receiwd
Key words: Nonlinear elliptic Leray-Schauder degree.
boundary
Olshausenstralje
1 February
value problem,
40-60,
D 2300 Kiel, W. Germany
1979)
critical
point
theory,
global
convex
reduction,
INTRODUCTION THE AIM
boundary
of this paper is the study of existence value problems of the type
and multiplicity
Lu(x) = p(.\‘,u),
questions
for nonlinear
elliptic
XEG
XEaG
u(x) = 0,
(DP)
where L is a uniformly elliptic and formally selfadjoint operator of second order and G is a sufficiently smooth, bounded domain in R".We have been motivated by recent papers of Ambrosetti and Mancini [ 1, 21 and Castro and Lazer [3] who reduce the Dirichlet problem (DP) to a finite dimensional one and then apply elementary critical point theory or finite dimensional degree theory. Our main object is to show that a reduction similar to Castro and Lazer’s one also works in the case where the reduced problem is an infinite dimensional one leading to some new results for (DP). More precisely we substitute the search of critical points of the energy functional I belonging to (DP) by the search of critical points of a coercive functional J defined on a closed subspace of g’(G). This method can be interpreted in the framework of Berger’s approach to variational problems using natural isoperimetric constraints [4] as will be indicated in Section 2. Whereas Castro and Lazer treat (DP) under assumptions, among other things roughly requiring that pu(x, u) is increasing when IuI is tending from zero to infinity, we are able to consider the dual case too in which pu(x, u) is decreasing when (~1 goes from zero to infinity. We obtain the same type of results namely existence or multiplicity theorems-the strongest ones if p(x, u) is odd in u. Furthermore Castro and Lazer are only concerned with nonresonance cases in which--somewhat simplifiedAk < y d holds for two successive
eigenvalues
p(x,u)/u
<
Ak+I
(*)
of
Lu(x)= Au(x), XEG u(x)= 0,
x~aG.
Using a recently developed sharp criterion of Ahmad et al. for the existence problems at resonance [5] we obtain results not requiring condition(*). 195 D
of solutions
of elliptic
796
KLAUS THEWS
Ambrosetti and Mancini [l ,2] study (DP) where p(x, u) = i,u + f(x, u), f‘(x, u) being uniformly bounded (except when k = 1). Our assumptions do not require boundedness of f(x, u) and therefore we can generalize some of their results. It is also possible to treat higher order variational Dirichlet problems by the same method as for instance is done in the author’s dissertation [6] in which most results of this paper are contained.
1. STATEMENT
We consider
the boundary
h(x) = -
OF
THE
MAIN
RESULTS
value problem
i
Di(Uij(X)DjU(X)) + c(x)u(x) = p(x, u(x)),
XEG
i,j=l
u(x) = 0,
x~c?G
where L is a uniformly elliptic differential operator of second order in a bounded with a smooth boundary 8G. The coefficients aij, aij = aji, are assumed to differentiable in G having Holder continuous first derivatives. Furthermore, continuous in G, c(x) 3 0 for all x E G and p be locally Holder continuous in G In addition we require: (i) There are positive
constants
(DP)
domain G c R" be continuously let c be Holder x R.
ci, c2 such that IP(X,Z)l d (‘1 + (.2lzl
for all x E G and all z E R; (ii) p E C’(G x R),and there are positive
constants
a, b such that
(p,(x, 41 d a + blz(“~ I for all x E G and all z E R where s is a fixed number n > 3 or 1 < s if n = 1,2. The eigenvalues
1 < s < (n + 2)/(n - 2) if
of the linear problem Lu(x) = AU(X), U(.Y)= 0,
are denoted as 0
satisfying
(I)
XEC.?G
each counted with multiplicity, E(Lk) is the eigenspace norm I/. II on i’(G) induced by the inner product G ~u,~D,uD~v SC,
u of (DP) is by definition
an element
(u,v) = holds for all v in i’(G). The next theorems contain
XEG
our main
sG
results
+ cuv
dx. >
u in h’(G)
such that
P(X, u(x,)v
concerning
classical
solutions
u of (DP) that
A reduction
method
for some nonlinear
means u E C’(G) n C(G) satisfying the equations h E C(G x R) we define H(x, z): = ji h(x, s)ds. THEOREM
Dirichlet
797
problems
in (DP) pointwise.
Corresponding
to a function
1. For p in (DP) we assume
(a) P& 4 < 2, + 1 for all x E G and u E R. (b) There is a constant M > 0 and hi, h, E C(G x R) such that (i., n + ki(X, 4)/n
d P(X, 4/u
d &+ 1 -
I+,
4)/u
holds for x E G and IuI 3 M. There is a positive constant K such that Ilz,(x, u)/ d K for i = 1,2 and all arguments. Furthermore let
s
H,(x, 4(x)) dx -+ co,
if+ E WA,),
II4 + cc
G
and
H,(x, $(x)) dx --f cc1 if II/ E -WI+ J,
/Ii4 -+ 3c‘.
sG (M) Under these conditions for all arguments. (p) In addition
(DP) has at least one solution.
The solution
is unique
if i,
< pu(x, u)
to (a) and (b) we assume: p(x, 0) = 0 for all x E G,
(c)
pu(x, 0) < AI for all x E G. Then, besides problem
the trivial
solution,
(DP) has a nontrivial Lw = pu(x, O)w, w = 0,
has only the trivial weak solution, (y) Let conditions
solution.
If the linear
G i3G
then there exist at least two nontrivial
(a) and (b) be satisfied
eigenvalue
solutions
for (DP).
and assume that
p(x, u) = - p(.~, - u) for all x E G,
UER
as well as Pu(% 0) < A+,+ 1 for all x E G, (I - k + 1 2 1). Then (DP) has at least k pairs of nontrivial
solutions.
Remarks (i) Conditions for h,, h, as demanded in (b) have been introduced by Ahmad et al. [5] in order to generalize the Landesman-Lazer conditions which are useful in resonance problems. The assumption on H,(x, z) as stated in (b) is satisfied in particular if H,(x, z) + co as IzI -9 W uniformly
in x E G. (Compare
with [13]).
798
KLAUSTHEWS
(ii) Part (fi) of Theorem 1 is a generalization of Theorem A in [3] where condition (b) is replaced by a stronger nonresonance-condition, in particular demanding that pu(x, u) < y < A,+ 1 holds for all arguments. In the same sense part (y) is a generalization of Theorem C in Castro and Lazer’s paper. Except for regularity assumptions and the use of only one linear eigenvalue problem to control the behaviour of p(x, u) at u = 0 and u = cc part (y) can be regarded as an extension of Theorem 1 in [8] due to the weakening of condition * mentioned in the introduction. (iii) Similar results as stated in Theorem 1 and Theorem 2 below are contained in Ambrosetti and Mancini [2] (compare with Theorems (3.3), (4.2), (5.4), (5.7) and remark (3.4)) but there the assumption is made that Ip(x, u) - I,ul d K’ holds for a fixed constant K’. Ambrosetti and Mancini use a global Ljapunoff-Schmidt an upper bound for p,,(x, u) which is not required in the next Theorem.
method
needing
THEOREM2. For p in (DP) we assume UER
p,,(x, u) > A, for all x E G,
(a’)
and (b) as in Theorem 1. (a) Under the assumptions stated above (DP) has at least one solution. (fi) In addition to (a’) and (b) we assume: p(x, 0) = 0 for all x E G, p&, 0) > A,, , for all x E G. Then. problem
besides
the trivial
solution,
(DP) has a nontrivial
Lw = p&x, O)w, w = 0,
solution.
If the linear
G f3G
has only the trivial weak solution, then there exist at least two nontrivial (y) Let conditions (a’) and (b) be satisfied and assume that p(x, u) = -p(x,
eigenvalue
-u) holds for x E G,
solutions
for (DP).
UER
as well as p,(x,O)> Then (DP) has at least k pairs of nontrivial
;1,+,forxEG. solutions.
Remark. Due to their special method of reduction Castro and Lazer do not obtain an analogue of Theorem 2. For the same reasons as before part (y) can be regarded as an extension of Theorem 2 in [S]. The proof of Theorems 1 and 2 will be accomplished by finding critical points of the energy functional I associated with (DP). TO this end we give an extension of Theorem 4 in [3] which is an abstract result about critical points of functionals on Hilbert spaces.
A reduction
2. CRITICAL
POINTS
OF
method
SADDLE
for some nonlinear
TYPE
Dirichlet
FUNCTIONALS
799
problems
ON
HILBERT
SPACES
First we recall some definitions. Let E be a Hilbert space with inner product (. , .), corresponding norm (/. /I and (. , .) as duality pairing between E and its dual space E*. The orthogonal complement of a subspace W c E in E is denoted by WI, the orthogonal sum of two subspaces r/; W c E by V@ !4! For z E E, r > 0, B,(z) denotes the open ball with center at z and radius r, B,(z) its closure. As usual, L(E) is the set of all bounded linear operators from E to E, GL(E) c L(E) is the subset of boundedly invertible ones. A mapping h: E + E is called compact if it is continuous and maps bounded sets into relatively compact sets. If A is a bounded, selfadjoint operator on E the essential spectrum o,(A) of A consists of all elements of the spectrum of A which are not isolated eigenvalues of finite multiplicity. A functional I: E + R is said to be coercive if lim Z(z) = cc. I/~l/+m A functional I: E -+ R is called weakly continuous if it maps weakly convergent sequences into convergent sequences. Denoting weak convergence in E by - we call I weakly lower semicontinuous if zn - z in E implies I(z) < lim inf I(z,). THEOREM 3. Let E be a real Hilbert space, f E C2(E, R) such that Vf: E --f E is a compact mapping. A is assumed to be a bounded self-adjoint operator on E such that 0 4 a,(A). Let E, be the space AT/c I/: where A is positive definite and T/c E + a subspace satisfying I/ = V, I:E-Risdefinedby
Z(z) = +(Az, z) + f(z). We assume (a) (i) 11T/is coercive. (ii) -I is coercive on v’, uniformly on bounded subsets of T/ (i.e. for all M > 0 and R > 0 there exists 6 > 0 such that Z(u + w) < -R if u E r/; w E V’ and llzlll< M, llwll>, 6). (iii) For all z E E and all w E V’ \ (0) we have (Z”(z)w, w) < 0. Then c = inf sup Z(u + w) is a critical value of I. If in addition (1”(z)u, u) > 0 for all z E E and all vcv KC.V u E I/\(O) then 1’possesses exactly one critical point. (b) Let conditions (iHiii) be satisfied as well as (iv) 1’(O)= 0 and there is a nontrivial subspace t c T/such that (Z”(O)& u) < 0
for all nonzero elements in V + VI. Then I has at least one nontrivial critical point. (c) Assume (iHiv) as well as : (v) I”(O), interpreted as a bounded linear operator from E to E, has a bounded inverse i.e. r’(0) E GL(E). Then I has at least two nontrivial critical points. (d) Let condition2 (iHiv) be satisfied, I an even functional, Z(0) = 0. Assume that for p from (iv) we have dim V 3 k for a natural number k. Then I has at least k pairs of nontrivial critical points.
KLAUS
800
THEWS
Remarks (i) Theorem 4 of Castro and Lazer yields essentially the same conclusions but requires dim V < co. On the other hand Theorem 4 deals with a more general class of functionals than our Theorem 3. (ii) The existence statement in part (a) which is rather straightforward can also be derived from Berger’s Theorem (6.3.13) in [4]. Berger’s multiplicity statements however use Morse theory and are only generic, furthermore Vf is assumed to be uniformly bounded and A to have a finite Morse index (Compare with Theorem (6.5.16) in [4]). (iii) The assumptions on A and f and condition (ii) imply that I/ has finite codimension Proof: 0 4 a,(A) implies that there is a constant m > 0 such that
in E,.
~Az, z) 3 rnllz11’for all z E E,. We have E + = I/ @ I/’ where J’/’ = 1/l n E +, According to (ii) there is a 6 > 0 such that I(v’) d - 1 for u’ E V’, I(v’I( 2 6. Suppose that dim V’ = co. Then there exists a sequence (0;) in V’ satisfying v’ - 0 in V’ (and, of course, then 0: - 0 in E) and 11 u’,)I = 6. A s is well known the compactness of ?j” implies that f is weakly continuous (Theorem 8.2 in [9]) and therefore f(uh) + 0. For n sufficiently large we have Z(u:) = &4vb, vk) + f(ui) which is a contradiction
to our conclusion
3 $rn@ > 0
from condition
(ii).
Proof of Theorem 3. (a) Let g,: I/’ + R be the coercive functional defined by g,(w) = - I(u + w) = -+((Au, 0) - $(Aw, w) - f (u + w). As VI = v’ @ E, @ E_ where E_ is the space on which A is negative definite and dim ( V’ + E,) -c co we conclude that g, is weakly lower semicontinuous. (Compare with example 8.2 in [9].) Therefore there is an element F(u) E V’ satisfying g,@(o)) = min g,(w). WEV
Condition (iii) implies the uniqueness ForrEI/;w,wl,wzEI/‘wehave
of W(u):
<&(w,) - SXWZ), Wl
-
w2)
1 =
GI:‘(w,+
z.(w,
-
w,,)(wl
-
wz),
~1
-
wz>
dT
s 0
=
s1-
(I”(0 + wz + +“, -
w,,)b”l - ~21,~1
-
~2) dT
0
3
a(~, wl,
w2)IIw1- ~~11’ with a(~,wl, w2) > 0
if w1 # w2 according to (iii). This relation implies and is a strictly monotone operator. We claim that W: V + l/i is a C’ mapping.
that &: I/’ -+ (V’)* can only have one zero
A
reduction
method
for some nonlinear
Dirichlet
problems
801
Proof: W(v)is the uniquely determined element w in 1/l for which (Z’(u + w), @) = 0 holds for all 6~E T/l. Denoting by R the orthogonal projection 71:E + 1/l this is equivalent to 7to VZ(u+ w) =O, To solve the equation 7~oVZ(u + w) = 0 for w we use the implicit function theorem. Let u0 E y w,, E 1/l be given such that 71oVZ(u, + WJ = 0. We define the mapping F:Vx
l/l+l/lby(u,w)+noVZ(u+w)
which is a C’ mapping. We want to show D,F(u,, wO)E GL(1/l) and compute
= d/dt(n: oVZ(u,, w,, + tw,), w,>l,zO
= (I”(u, + WO)Wl,WJ = (7coZ”(V, + W&WI,w,) where wl, w2 E 1/‘-and of course Z”(uO+ wO)is regarded as an element of L(E). In view of Al/ c I/ and AV’ c 1/* we get 7t oZ”(U, + WJ = T + 7cof’@,
+ WJ
where T = A(V’: 1/l + 1/l. TE L(I/‘) is a Fredholm operator of index 0 and z of”(u, + WJ is a compact endomorphism of V as f”(u, + wO)E L(E) is compact as a derivative of a compact mapping (compare with Theorem 4.1 in [lo]). Therefore D,F(u,, WJ is Fredholm of index 0 so that it suffices to show: KernD,F(u,, wO)= (0). But this follows from (D,F(u,, wO)wI, wl) = (Z”(u,, wo)wl, w,) < 0 for w1 # 0 which is a consequence of (iii). So our claim follows from the implicit function theorem. After these preparations we are able to reduce the search for critical points of I to that of a simpler functional G. Definition. Let G: V -+
R be given by G(u) = Z(u + W(u)).
LEMMA 1. z E E is a critical point of Z if and only if z = u + W(v)and u is a critical point of G. Proof: Each critical point z of I has the form 2 = u + W(u).The assertion now follows from the formulae (G’(u), 6) = (d/dt)Z(u + tv” -t W(u + tv”))l,,o = (Z’(u + W(u)),5 + (W)yu)iQ =
where u, 6 E I/: The last equation implies that G E C2( K R) and :
(G"(u)u1,u2) = (I"@+ E(u,)(q + W'(u)u,), u2). We note that G is coercive as G(u) 2 Z(u) + 00. To prove the first part of (a) it is enough to show that G is weakly lower semicontin!%z ?m V because then G achieves its infimum on V which is a critical value of G and also of I. For preparation we prove:
KLAUS THEWS
802 LEMMA
2. The mapping
W: V + I/’ is bounded
on bounded
sets.
Proof. Let K > 0 be given. As I is bounded on bounded sets there is a constant L > 0 such that Z(u)3 -LforvEX I/u// < K.D ue t o condition (ii) there is a constant 6 > 0 with the property: For w E V1, u E V, llull < K, llwll 3 6 it follows I(u+w)d As Z(u + W(u)) = sup Z(u + w) we conclude wcv’ LEMMA 3. G:
-2L.
[Iw(u)[~< 6.
V + R is weakly lower semicontinuous.
Proof: Let rc’: E + V denote the orthogonal projection. For V,G which is the gradient with respect to the inner product on V induced by E we have:
of G
V,G(u) = 7~’o VZ(u + W(u)) = Au + rr’o Vf(u + W(u)), As G(u) = $4u,
v) + $(AW(u), W(u)) + f(u + W(u)) we conclude 71’0 Vf(u + W(u)) = V,($4w(u),
W(u)) + f(u + W(u))).
mapping and bounded The mapping u -+ rc’o V’(u + W(u)) is compact because W is a continuous on bounded sets and Vf is compact. Therefore the associated potential is weakly continuous. Now G is weakly lower semicontinuous as a sum of weakly continuous functional and a weakly lower semicontinuous one. The proof of the first part of assertion (a) is now complete. Concerning the second part, under the additional hypothesis, we remark that u + I(u + w) is a strictly convex function for fixed w E V1. The uniqueness result is then a consequence from Lemma 1 in [ 111. (b) Condition (iv) implies G’(0) = 0. It suffices to show that in 0 no local minimum achieved. For vi E V\(O) we have
for G is
(G”(O)o,, vi> = U”(O)(u, + W’(O)u,), vi> = (I”(O)(U, + w’(O)u,), u, + W’(O)u,) the last equation
being valid because (I’(u,
and all w E I/‘. Choosing
in particular
+ W(u,)), w) = 0 for all vi E V u1 E v\(O) we get
01 + (3ol)(a,) E (v” + VI)\(O) andlinally (G”(O)u,, vi) < 0. Therefore in 0 there is no local minimum of G and for the critical critical value c we have u,, # 0 because rc’u, # 0.
point
a0 belonging
to the
(c) The arguments here are based on two topological results the proof of which will be given in the next section. First we note that assumption (v) implies G”(0) E GL( V). Proof. An analoguous reasoning as in the case of D,F(u,, wO) shows that G”(0) is Fredholm index 0. So it suffices to prove that the kernel of G”(0) is trivial.
of
A reduction
method
for some nonlinear
Dirichlet
problems
803
Assume that v1 E V is such that 0 = (G’(O)v,, u) for all v E I/: This implies 0 = (Z”(O)(v, + W’(O)v,),U)for all u E I/ and in view of 0 = (I”(O)(v, + W’(O)v,),w) for all w E I/’ leads to 0 = Z”(O)(v, + W’(O)v,).Finally we get 0 = v1 + W’(O)v, and
0 = vl.
As shown above 0 and II,, = rr’r+,are two different zeroes of VG. We can assume that u0 is an isolated zero of VG and that VG has no zeroes the norms of which tend to infinity because otherwise(c) would be trivial. We now choose ((v,, II,)): = (Au,, vJ as a new inner product on 5 equivalent to the one induced by E and note that for the gradient of G with respect to ((.,.)), VG, we have ~G(v) = u + K(v) where K: V + V is a compact mapping. Now the assertion follows from properties of the Leray-Schauder degree. As G”(O)EGL( V) for sufficiently small I > 0 we have deg (VG, B,(O),0) = deg (@G)‘(O),B,(O),0) = -&1. From Theorem 5. in the next section we get for the local degree of VG at v,, which is a local minimum of G and an isolated critical point : deg(vG, B,(Q), 0) = 1 for small s > 0,
(*)
and for large M we obtain due to Theorem 4. deg (?G, B,(O), 0) = 1.
(**)
According to the relation deg (?G, B&J, 0) + deg (?G, B,(O), 0) # 1 ?G must have another zero in B,(O). This proves (c). (d) For the proof of this part we apply Clark’s Theorem 11 from [12] to our functional G. Before stating Theorem 11 we need a definition: A real valued Cl-functional f on a real Banach space E such that f(0) = 0 is said to satisfy (C) if every bounded sequence (x,) c E with the properties f(x,,) < 0, f(x,) bounded below and f’(x,) + 0 contains a convergent subsequence. The assertion of Clark’s Theorem 11 is : Let E be a real Banach space and f be an even, real valued, Cl-function on E. Suppose further that f satisfies condition (C), f(0) = 0, f is bounded below, f(x) 3 0 for large IIxI[,and f(x) = q(x) + o(IIx(12)as I/x(1-+ 0, wh ere 4 is a quadratic form of index k. Then f has at least k pairs of nontrivial critical points. We have to verify that G satisfies the assumptions above, As Z is even G is even and G(0) = 0. From the formula VG = id + K we conclude that G is bounded from below and that it satisfies condition (c):
KLAUS THEWS
804
Let (u,) c V be a bounded sequence such that VG(u,) --*0. As K is compact there is a subsequence u,,, such that K(c,,) converges, K(u,,) --) u in V: Since u,! + K(ti,,) 1_m + 0 we obtain u,, + - u. Lastly, from the hypothesis dim P > k and the expression for G”(0) we conclude that the dimension of space where G”(0) is negative definite is greater or equal k. The application of Clark’s theorem yields k pairs of critical points for G and therefore Z has at least k pairs of nontrivial critical points. Remark. Berger [4] calls a submanifold A4 of a Hilbert space H a natural isoperimetric constraint for a Cl-functional J on H if the following conditions are satisfied:
(i) c = inf J(u) is attained by an element f E M. UEM
(ii) For each ii E M n J-‘(c) if follows J’(U) = 0. (iii) Each critical point of J(u) lies on M. Using this notation it is easy to see that in our case where H = E and .Z = Z M : = {z E E, VZ(z) I V’} is a natural isoperimetric constraint for I. The described reduction method then consists of looking at Z only on M and using a parametrization for M: M = {(u + W(u)),u E V}, $:V+M u + (u + 5(u))
is a parametrization
3. THE
of M. Then for G we have the representation
LERAY-SCHAUDER
DEGREE
OF
SOME
SPECIAL
G = lo $.
GRADIENT
OPERATORS
This section is devoted to the proof of generalizations of two well known facts concerning the Brouwer degree of the gradients of some functionals, namely Theorem 4. which immediately implies formula (**) in the proof of Theorem 3. (c) and Theorem 5. which yields formula (“). THEOREM
4. Let H be a real Hilbert space and G E C’(H, R) be given by G(w) = $11~11”-f(w)>
where V’: H -+ H is a compact mapping. Furthermore G is
non-degenerate,
WGH
we assume that
i.e. there is a constant R > 0 such that
lim
G(w) = cc and that
liwli-+m
I[wI[>, R implies VG(w) # 0. Then, for sufficiently large M, we have deg (VG, BJO), 0) = 1. Remark. In the special case H = R” this theorem is just the assertion of Krasnosel’skii’s Lemma (6.5) in [13]. We make essential use of some ideas of Krasnosel’skii’s proof.
Next we formulate a local analogue of Theorem 4. :
805
A reduction method for some nonlinear Dirichlet problems THEOREM 5. Let H be a real Hilbert
space, y E 23, R an open neighborhood
of y and
G E C’(Q R) where G(w) = ill w(I2 - f(w) and V’: Q + His a compact Assume that y is a local minimum for G and an isolated Then, for sufticiently small r > 0, we have
mapping.
critical point.
deg (VG, B,(Y), 0) = 1. Remark. In the case G E C2(S& R), an assumption which is satisfied in our application, Rabinowitz, for instance, gives a proof of this assertion (see [ 141). But as his proof seems to be complicated we give another proof which is completely analogous to that of Theorem 4. Proof of Theorem 4. We begin with a preliminary LEMMA1. Let R > 0 be given. Then there is a separable Hilbert space V c H satisfying Vf (B,J c I/: Let (y,), E N be a complete orthonormal system for I/ and x,,: H + span {y,, . . . , y,} the orthogonal projection. Then for each E > 0 there is m(c) E N such that n 3 m(E) implies /IVf (z) - 71,o Vf (z)I/ d F
for all z E BR.
Proof. N: = Vf(B,) c H is compact and separable as {w,, n E N) be a countable dense subset of N. Then I/: = such that Vf (B,) c v Now assume that the second assertion is wrong. Then there are E > 0, sequences mj + 00 and zj E BR such IIvf(zj)
Using the compactness
-
n,jvf(zj)II
a compact metric space. Let M = . span M is a separable Hilbert space
that
3 ”
of Vf we can assume Vf (zj)j_,
-+ u E v.
Then ‘f =
v
+
('j)
-
%ljvf
(Vf(Zj) - u) - 71JVf(Zj)
('j)
- u) - 7&u
= (id - 7&J(u) + (Vf(Zj) - u) - 7c,,(Vf(Zj) - u). All terms on the right hand side converge
to 0 which leads to a contradiction
LEMMA 2. G satisfies
bounded
the Palais-Smale condition: Each sequence and VG(w,) + 0 contains a convergent subsequence.
to our assumption.
(w,) c H such that (G(w,)) is
Proof: (G(w,)) is . b ounded from above and therefore (w,,) is bounded taking into account the coercivity of G. Using the compactness of Vf we find a subsequence Vf (wnr) + w E H. Since Vf (w,,) we conclude wnk + w. As a consequence we note that for any closed VG(w,J = w,~ subset M c H on which VG has no zeroes there is a positive constant 6, > 0 such that IIVG(w)ll 2 6, for all w E M. After these preparations
we come to the actual proof: As G is nondegenerate
there is a constant
KLAUS THEWS
806
R, > 0 such that VG(w) # 0 for ljwli 3 M,:
R,.As G IS . b ounded on bounded sets we define =
sup G(w) < W. llwll
Using the coercivity of G we find that R,: = {w E H, G(w) d M, + 1) is bounded. Now we choose p > 0 and R, > p such that BR, _p I R, and define M, : = sup G(w) + 1 as well as I/WI,
let
< M,)
R2 begiven with the property Q2, = BRZ.
Since VG does not vanish
on {WEE&M, LQ: =
+ + < G(w) < M,f
Mo+ l;:nfilw,-_M, IIVGCw)II
is strictly greater than zero. Let K: = ,,w”pzR_tlVG(w)ll < co. 7 NOW we
apply Lemma
1, taking
R = R, and F = min{(a,/2),
En: = span (yl,. . . , y.} E, is equipped
with the inner product
and G,: E, + R induced
by
(c(i/4K)}. For n >, m(s) we define
by G,(w) = G(w)
for w E E,.
E.As usual, defining V,G,(u) by the requirement for all II E E,,
(V,G,(u), u) = (Gn(u), u)
we obtain V,G,(u) = rr, o VG(u) = u - n, o V’(u) for all u E E,.Define B,:
=E, n BR,. The essential step is now to compute the Brouwer degree of
V,G,. ASSERTION.
deg (V,G,, B,, 0) = 1 for n 2 m(c).
Proof: There is a locally Lipschitz
continuous
vector field H,: BRZ n
11H,(u) - rc, 0 VS(u)ll < F’ where E’: = min {a,/4, c$4H}. This implies the ordinary differential equation
E, -+E,satisfying
VuEE,ni?Rz
deg (II,, B,, 0) = deg (V, G,, B, 0). Now we consider
1 = -H,(x) (*)
x(0) = x0 in the domain
int (Q,)
n E,.Let x(t) be a solution
such that
M, + $ d G(x(t)) < M,.
A
reduction
method
for some nonlinear
Dirichlet
problems
Then we compute = -(VG(x(t)),
(dldr) G@(r)) = (VG(x(t)), i(r))
~,(x(t)))
< - a;/2
(**)
using the definitions of 01~)K, E and E’. Therefore, trajectories starting in int (0,) n E, remain there and the solutions are defined for all positive times. Let U(t) be the translation operator associated with (*) i.e. U(t)x is the solution of (*) with initial value x at the time t. All elements x E a!, c int(n,) n En are points of irreversibility for (*) i.e. for all t > 0 we have x # U(t)x, x E adl,. This follows because G is strictly decreasing along the solutions x(t) if x(t) E RI\&,. As H, has no zeroes on 8B, Lemma (6.1) in [13] gives the information: deg (H,, B,, 0) = deg (id Now relation
(**) implies that a trajectory ZE
and remains
starting
U(t), I?,, 0).
in 8B, is in R, at a certain
1
2&f, -of,+ 1))
6 o
there in future. Let T > 2(M,
2 MO
- (MO + l))/a: be a fixed number.
U(T)x E Cl0 n E, c B,, n E, The homotopy shows that
time
H(s, x) = x - sU(T)x,
O
deg (id -
Then
for all x E 8B,. XEB,
which
has no zeroes
on 8,
U(T), I?,, 0) = 1.
This proves the assertion above. Finally the definition of the Leray-Schauder degree, [15], taking X, : =E,, F, : = nn,OV'leads to the formula
as for instance
given in section
16 of
deg (id - f, BR,, 0) = deg (I,,, - 71,o Vf, B,,, 0)
which proves Theorem
4.
Proof of Theorem 5. Without loss of generality we assume y = 0, G(0) = 0 and y = 0 being a strict local minimum. We choose R, > 0 such that llwll G Roy We need a preliminary
w # 0
imply VG(w) # 0, G(w) > 0.
lemma:
LEMMA.For 0 < R, d R, we have inf
G(w) > 0.
R,QIIw~lQR Proof. Assume that the assertion is wrong. Then there are w,, E H, R, d llw,/I d R, satisfying G(w,,) -+ 0. As BRo is weakly sequentially compact we assume w, - w, w E BRI,. Consequently
11 w II < lim inf IIw, II and f(w,) + f(w).
KLAUS THEWS
808
From
G(w,) = tilwnII 2 - f(w,) + 0 we conclude G(w) = +l/wll’ -f(w)
411w,Il 2 + f(w). Therefore
d +lim IIw,(12 -f(w)
= 0.
This implies G(w) = 0 and w = 0. On the other hand, due to R, 6 IIwnll,we find llw,,1123 $RT > 0
fiw) 3 $liminf
and G(w) = G(0) d -iRf which is a contradiction. Now we come to the actual proof: Let0 < c(~: = inf G(w) be chosen. Define CI, : = {w E BRo, G(w) 6 a,/2} and choose 0~
R, <
II4=Ro R', such that BR, c BRi c n,. Then define cl, : =
inf G(w)>0 andQ,:= RI< II4 SRo deg (VG, B,,, 0) = 1 is proven completely analogous
4. PROOFS
as in the preceding
1 AND
OF THEOREMS
First we collect some standard facts concerning Let E: = WA, 2(G), f: E + R be defined as f(u) =
{w E BRo, G(w) < cr,/2}. Finally
P(x, u(x)) dx
the functional
where P(x, z) =
associated
for u E E. Due to our assumptions of (DP) and also classical
proof.
2
analytic
treatment
of (DP).
’ p(x, t) dt. s 0
sG The energy functional
the fact that
with (DP) is defined by
(i) and (ii) we have I E C2(E, R). Critical points of I are weak solutions solutions by known regularity results. For u, u E E
= IG (;
aijDiuDjv + c(x)m)
dx - jG p(x, u(x))v(x) dx holds.
For u, v, w E Ewe have pu(x, u(x))v(x)w(x) dx.
(Z”(u)u, w) = (u, w) sG As already mentioned elliptic operator L.
in Section 1. E will be equipped
yf: E ---fE f :E + R The linear eigenvalue
with the inner product
is compact
(. , .) arising from the
and
weakly continuous.
problem Lu(x) = Au(x), u(x) = 0,
xeG xedG
VA
A reduction
method
for some nonlinear
Dirichlet
809
problems
possesses an unbounded sequence of eigenvalues 0 < I, < I, < i3 . . . + co and a corresponding system (Q,) of eigenfunctions which is a complete orthonormal system in (E, (. , .)). For u E span {@I,. . ., @‘l1 we have the inequality
s s
u’(x) dx
(4 4 G 1, and for w E (span {aI,.
G
. . , rI$))’ we get
(w,w)3 Al+1 w’(x) dx. G
For proofs of these statements
and additional
references
Proof of Theorem 2. Our aim is the application We choose E = WA* *(G),
we refer to [6,16].
of Theorem
3 to find critical
points
of 1.
Z(u) = 411q2 - f(4 where
f(u) =
s
P(x, 4x)) dx
G
and
I/: = (span {Q,,, . . . , CD[})*, A: =id,. Then 1/l = span {aI,. Now we show that the assumptions Verification of(i). Using the right inequality f(u) =
of Theorem
in hypothesis
3. are satisfied.
(b) we obtain
P(x, u(x)) dx d $ sG
. . , ml}.
A,, 1u2(x) dx -
Z-Z,(x, u(x)) dx + const. s G
s
and therefore Z(u) > fllull’
- fA,+,
u’(x) dx + I’G
where the appearing constants v’ E E(A, + ,) and v” E (Et,$+ ,))‘.
H,(x, u(x)) dx - const. sG
are independent
of u. Let v E V be given,
Using that
11 v” 11’2
u”(x)~ dx
p
for a constant
sG P ’
A,+, and
that the system (a,) is also &-orthogonal Z(v) 2
(1 - %) IIv”l12
we conclude:
+ JGH2(x,v(x))dx
- const.
v = v’ + v” where
810
KLAUSTHEWS
As 1h,(x, u) 1is uniformly bounded and L’(G) continuously embedded in L*(G) it follows
/u”(x)1dx - const.
H,(x, u’(x)) dx - K
+
sG
sG
H,(x, u’(x)) dx - const.
+
sG The last inequality together with the hypothesis concerning H, yields the coercivity of I on I/. Verification of (ii). The left inequality in hypothesis (b) leads to I(u) d
($jGALu’(x) dx + jG H, (x, u(x)) dx -
+lluII2 -
cons,.).
Splitting
u=u+w,
v,
uE
WE
VI, w”
w’ E JW, ),
w = w’ + w“,
E
E(A, )’
we obtain:
44 G 3llull’ + lIw’l12 + IIw”l12) -
$A, G (u’(x)+ w’(x)~+ w”(x)‘) dx
-
s
s
H,(x, u(x)) dx + const.
G
u2(x)dx
< +Ilull” - +A, sG
w”(x)~dx
+ $llw”l12 - +A,
sG c -
J G
<
r H,(x, w’(x)) dx + K G Iv(x) + w”(x)Idx + const. J
fllull’ - $2,
s
u’(x) dx + K G 1u(x)] dx
G
+
-
w"(x)~
$tv -4)
s
s
dx + K G 1w”(x)1 dx
sG
H,(x, w’(x)) dx + const.
G
s
A reduction
method
for some nonlinear
Dirichlet
811
problems
Here Yis a constant satisfying v < ir. We have used that the eigenfunctions Oj are C-orthogonal and that (h,(x, u) ( d K. The assumption on H, and the last formula then imply condition (ii). Verification of (iii). For z E E, w E V”\(O) we get (Z”(X) w, w) = (w, w) -
s
JP,(x,z(x))w’(x)dx G
(A, - p,(x, 4x)))w’(x)dx < 0.
<
G
Now we have shown that assumptions (a’) and (b) of Theorem Theorem 3. This proves Theorem 2, or). Next we show that condition
(iv) is a consequence
Verification of (iv) p(x, 0) = 0 implies Z’(0) = 0. Choose
2, imply conditions
of the first hypothesis
(iHiii) from
in (/I).
v : = R. (I$+ 1. Then for u E p + 1/l we have
(Z”(O) u, u) = (u, u) -
pu(x, 0) u2(x) dx sG
d
((4,
J
-
P,,(x,
0))
u’(x)
dx
<
0,
s G
ifu # 0. Part (b) of Theorem additional assumption
3 now proves the first assertion in (j?) implies hypothesis (v).
of@). The second assertion
is true since the
Verification of(v). Z”(0) E L(E) is Fredholm of index 0. Therefore it is enough to show that Kern Z”(0) = that there is u E E such that Z”(0) u = 0. Then (Z”(0) v, u) = 0 for all u E E and (u, u) u(x) u(x) dx = 0 for all u E E which means that u is a weak solution of
01. Assume G pU(x, 0) x
Lw = pU(x, 0) w, G aG.
w =o, But then the additional
hypothesis
shows that v = 0.
Part (c) of Theorem 3 then yields the second assertion of@). Finally, to prove part (y), we remark that, under the assumptions of(y), Z(0) = 0 and Z is even since P(x, z) is even in z. For t we choose span {QL+r, . . . , QD, +,} and notice that for u E v + I/’ = span{@,, . . . , @,I+ k} the following relation holds: (Z”(0)
u, u)
= (u, u) -
p,(x,
0) u2(x) dx
sG -
ifu # 0.
p.(x,
0)) u”(x)
dx < 0
KLAUS THEWS
812
Part (y) now is a consequence of assertion (d) in Theorem 3. This completes the proof of Theorem 2. Finally we indicate how Theorem 3 can be used to prove Theorem 1. As the details are similar to the proof above we shall be sketchy here. Proof of Theorem 1. This time, with the aid of Theorem 3, we are looking for critical points of -I on E = w:,‘(G). We define V = span{QI, . . . , (I+} and the operator A is defined by Au: = v - w
where for
UGE
we set u = v + w, v E V, w E 1/l. Then
44 =
=
-(illull’ - JG p(x,U(XW) -+llwll’ + +I~~’ +
(j-p(x, u(.y))dx- Ilull’). G
We shall briefly mention how the various assumptions of Theorem 3 can be verified in this situation. Conditions (i) and (ii) follow from hypothesis (b) using similar arguments as before. Condition (iii). For z E E, w E V”\(O) it follows: (I”(z) w, w) = -(w, w) +
s G
s
PAX,z(x))w’(x)dx
Q G(PAX, 4~)) -
4 +,) w2(x) dx < 0
in view of hypothesis (a). This proves the first assertion of part (a). Quite similar reasoning shows that the additional hypothesis in (a) implies that (Z”(z) v, v) > 0
for all
z E E, v E V\(O).
From this the uniqueness of the solution can be deduced. Condition (iv). This is satisfied in view of (c) with t:=
R.Q,.
Condition (v). This assumption is a consequence of the additional hypothesis in (p). Now (8) is a consequence of parts (b) and (c) of Theorem 3. Condition (vi). Choosing I/: = span{@,_,+ 1, _. . , CDI ) (vi) is satisfied under the assumptions of (y). Assertion (y) is then a consequence of part (d) from Theorem 3. This completes the proof of Theorem 1.
A reduction
method
for some nonlinear
Dirichlet
problems
813
REFERENCES 1. AMBROSETTIA. & MANCIN~ G., Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance. J. d$$ Eqns 28, 220-245 (1978). 2. AMBROSETTIA. & MANCINI G., Theorems of existence and multiplicity for nonlinear elliptic problems with noninvertible linear part. Ann. Scuola Norm. Sup. Piss 5, 15-28 (1978). 3. CASTRO A. & LAZER A. C., Critical point theory and the number of solutions of a nonlinear Dirichlet problem. Preprint. 4. BERGER M. S., Nonlinearity and Functional Analysis. Academic Press, New York (1977). 5. AHMAD S., LAZER A. C. & PAUL J., Elementary critical point theory and perturbations of elliptic boundary value problems at resonance. Indiana Univ. Math. J. 25, 933-944 (1976). 6. THEWS K., Untere Schranken fiir die Anzahl von Liisungen einer Klasse nichtlinearer Dirichletprobleme. Dissertation, Ruhr-UniversitLt Bochum (1978). 7. RABINOWITZ P., Some minimax theorems and applications to nonlinear partlal differential equations. In: Nonlinear Analvsis. a volume dedicated to E. H. Rot&. edited by Cesari. Kannan and Weinberaer, Academic Press (1978). Math. Z. 163, i63-175 8. THE& k., Multiple solutions for elliptic boundary value problems with odd nonlinearities. (1978). 9. VAINBERG M., Variational Methods for the Study oj’Nonlinear Operators. Holden Day, San Francisco (1964). New York 10. KRASNOSEL’SKII M., Topological Methods in the Theory of Nonlinear Integral Equations. MacMillan, (1964). 11. LAZER A. C., LANDESMANE. & MEYERSD., On saddle point problems in the calculus of variations, the Ritz algorithm, and monotone convergence. J. math. Analysis Applic., §3, 594-614 (1975). theory. Indiana Unio. Math. J. 22, 65-74 (1972). 12. CLARK D., A variant of the Lusternik-Schnirelman of translation along the trajectories of differential equations. Translations oj 13. KRASNOSEL’SKII M., The operator Mathematical Monographs, AMS (1968). 14. RA~~NOWITZ P., A note on topological degree for potential operators. J. math. Analysis Applic. 51, 483492 (1975). (1974). 15. DEIMLING K., Nichtlineare Gleichungen und Abbildungsgrade. Springer Hochschultext methods in critical point theory and applications. J. funct. 16. AMBROSETTI A. & RABINOWITZ P., Dual variational Analysis 14, 349-381 (1973).
Non/mm Analysis, Theory, Merhods & Appkofmns, Vol. 3, No 6, pp. 795-813 0 Pergamon Press Ltd. 1979 Prmted m Great Bntam
A REDUCTION
METHOD
FOR SOME NONLINEAR PROBLEMS
1101-0795 SOZ.oO/O
DIRICHLET
KLAUS THEWS Mathematisches
Seminar
der Christian
Albrechts-Universitlt, (Receiwd
Key words: Nonlinear elliptic Leray-Schauder degree.
boundary
Olshausenstralje
1 February
value problem,
40-60,
D 2300 Kiel, W. Germany
1979)
critical
point
theory,
global
convex
reduction,
INTRODUCTION THE AIM
boundary
of this paper is the study of existence value problems of the type
and multiplicity
Lu(x) = p(.\‘,u),
questions
for nonlinear
elliptic
XEG
XEaG
u(x) = 0,
(DP)
where L is a uniformly elliptic and formally selfadjoint operator of second order and G is a sufficiently smooth, bounded domain in R".We have been motivated by recent papers of Ambrosetti and Mancini [ 1, 21 and Castro and Lazer [3] who reduce the Dirichlet problem (DP) to a finite dimensional one and then apply elementary critical point theory or finite dimensional degree theory. Our main object is to show that a reduction similar to Castro and Lazer’s one also works in the case where the reduced problem is an infinite dimensional one leading to some new results for (DP). More precisely we substitute the search of critical points of the energy functional I belonging to (DP) by the search of critical points of a coercive functional J defined on a closed subspace of g’(G). This method can be interpreted in the framework of Berger’s approach to variational problems using natural isoperimetric constraints [4] as will be indicated in Section 2. Whereas Castro and Lazer treat (DP) under assumptions, among other things roughly requiring that pu(x, u) is increasing when IuI is tending from zero to infinity, we are able to consider the dual case too in which pu(x, u) is decreasing when (~1 goes from zero to infinity. We obtain the same type of results namely existence or multiplicity theorems-the strongest ones if p(x, u) is odd in u. Furthermore Castro and Lazer are only concerned with nonresonance cases in which--somewhat simplifiedAk < y d holds for two successive
eigenvalues
p(x,u)/u
<
Ak+I
(*)
of
Lu(x)= Au(x), XEG u(x)= 0,
x~aG.
Using a recently developed sharp criterion of Ahmad et al. for the existence problems at resonance [5] we obtain results not requiring condition(*). 195 D
of solutions
of elliptic
796
KLAUS THEWS
Ambrosetti and Mancini [l ,2] study (DP) where p(x, u) = i,u + f(x, u), f‘(x, u) being uniformly bounded (except when k = 1). Our assumptions do not require boundedness of f(x, u) and therefore we can generalize some of their results. It is also possible to treat higher order variational Dirichlet problems by the same method as for instance is done in the author’s dissertation [6] in which most results of this paper are contained.
1. STATEMENT
We consider
the boundary
h(x) = -
OF
THE
MAIN
RESULTS
value problem
i
Di(Uij(X)DjU(X)) + c(x)u(x) = p(x, u(x)),
XEG
i,j=l
u(x) = 0,
x~c?G
where L is a uniformly elliptic differential operator of second order in a bounded with a smooth boundary 8G. The coefficients aij, aij = aji, are assumed to differentiable in G having Holder continuous first derivatives. Furthermore, continuous in G, c(x) 3 0 for all x E G and p be locally Holder continuous in G In addition we require: (i) There are positive
constants
(DP)
domain G c R" be continuously let c be Holder x R.
ci, c2 such that IP(X,Z)l d (‘1 + (.2lzl
for all x E G and all z E R; (ii) p E C’(G x R),and there are positive
constants
a, b such that
(p,(x, 41 d a + blz(“~ I for all x E G and all z E R where s is a fixed number n > 3 or 1 < s if n = 1,2. The eigenvalues
1 < s < (n + 2)/(n - 2) if
of the linear problem Lu(x) = AU(X), U(.Y)= 0,
are denoted as 0
satisfying
(I)
XEC.?G
each counted with multiplicity, E(Lk) is the eigenspace norm I/. II on i’(G) induced by the inner product G ~u,~D,uD~v SC,
u of (DP) is by definition
an element
(u,v) = holds for all v in i’(G). The next theorems contain
XEG
our main
sG
results
+ cuv
dx. >
u in h’(G)
such that
P(X, u(x,)v
concerning
classical
solutions
u of (DP) that
A reduction
method
for some nonlinear
means u E C’(G) n C(G) satisfying the equations h E C(G x R) we define H(x, z): = ji h(x, s)ds. THEOREM
Dirichlet
797
problems
in (DP) pointwise.
Corresponding
to a function
1. For p in (DP) we assume
(a) P& 4 < 2, + 1 for all x E G and u E R. (b) There is a constant M > 0 and hi, h, E C(G x R) such that (i., n + ki(X, 4)/n
d P(X, 4/u
d &+ 1 -
I+,
4)/u
holds for x E G and IuI 3 M. There is a positive constant K such that Ilz,(x, u)/ d K for i = 1,2 and all arguments. Furthermore let
s
H,(x, 4(x)) dx -+ co,
if+ E WA,),
II4 + cc
G
and
H,(x, $(x)) dx --f cc1 if II/ E -WI+ J,
/Ii4 -+ 3c‘.
sG (M) Under these conditions for all arguments. (p) In addition
(DP) has at least one solution.
The solution
is unique
if i,
< pu(x, u)
to (a) and (b) we assume: p(x, 0) = 0 for all x E G,
(c)
pu(x, 0) < AI for all x E G. Then, besides problem
the trivial
solution,
(DP) has a nontrivial Lw = pu(x, O)w, w = 0,
has only the trivial weak solution, (y) Let conditions
solution.
If the linear
G i3G
then there exist at least two nontrivial
(a) and (b) be satisfied
eigenvalue
solutions
for (DP).
and assume that
p(x, u) = - p(.~, - u) for all x E G,
UER
as well as Pu(% 0) < A+,+ 1 for all x E G, (I - k + 1 2 1). Then (DP) has at least k pairs of nontrivial
solutions.
Remarks (i) Conditions for h,, h, as demanded in (b) have been introduced by Ahmad et al. [5] in order to generalize the Landesman-Lazer conditions which are useful in resonance problems. The assumption on H,(x, z) as stated in (b) is satisfied in particular if H,(x, z) + co as IzI -9 W uniformly
in x E G. (Compare
with [13]).
798
KLAUSTHEWS
(ii) Part (fi) of Theorem 1 is a generalization of Theorem A in [3] where condition (b) is replaced by a stronger nonresonance-condition, in particular demanding that pu(x, u) < y < A,+ 1 holds for all arguments. In the same sense part (y) is a generalization of Theorem C in Castro and Lazer’s paper. Except for regularity assumptions and the use of only one linear eigenvalue problem to control the behaviour of p(x, u) at u = 0 and u = cc part (y) can be regarded as an extension of Theorem 1 in [8] due to the weakening of condition * mentioned in the introduction. (iii) Similar results as stated in Theorem 1 and Theorem 2 below are contained in Ambrosetti and Mancini [2] (compare with Theorems (3.3), (4.2), (5.4), (5.7) and remark (3.4)) but there the assumption is made that Ip(x, u) - I,ul d K’ holds for a fixed constant K’. Ambrosetti and Mancini use a global Ljapunoff-Schmidt an upper bound for p,,(x, u) which is not required in the next Theorem.
method
needing
THEOREM2. For p in (DP) we assume UER
p,,(x, u) > A, for all x E G,
(a’)
and (b) as in Theorem 1. (a) Under the assumptions stated above (DP) has at least one solution. (fi) In addition to (a’) and (b) we assume: p(x, 0) = 0 for all x E G, p&, 0) > A,, , for all x E G. Then. problem
besides
the trivial
solution,
(DP) has a nontrivial
Lw = p&x, O)w, w = 0,
solution.
If the linear
G f3G
has only the trivial weak solution, then there exist at least two nontrivial (y) Let conditions (a’) and (b) be satisfied and assume that p(x, u) = -p(x,
eigenvalue
-u) holds for x E G,
solutions
for (DP).
UER
as well as p,(x,O)> Then (DP) has at least k pairs of nontrivial
;1,+,forxEG. solutions.
Remark. Due to their special method of reduction Castro and Lazer do not obtain an analogue of Theorem 2. For the same reasons as before part (y) can be regarded as an extension of Theorem 2 in [S]. The proof of Theorems 1 and 2 will be accomplished by finding critical points of the energy functional I associated with (DP). TO this end we give an extension of Theorem 4 in [3] which is an abstract result about critical points of functionals on Hilbert spaces.
A reduction
2. CRITICAL
POINTS
OF
method
SADDLE
for some nonlinear
TYPE
Dirichlet
FUNCTIONALS
799
problems
ON
HILBERT
SPACES
First we recall some definitions. Let E be a Hilbert space with inner product (. , .), corresponding norm (/. /I and (. , .) as duality pairing between E and its dual space E*. The orthogonal complement of a subspace W c E in E is denoted by WI, the orthogonal sum of two subspaces r/; W c E by V@ !4! For z E E, r > 0, B,(z) denotes the open ball with center at z and radius r, B,(z) its closure. As usual, L(E) is the set of all bounded linear operators from E to E, GL(E) c L(E) is the subset of boundedly invertible ones. A mapping h: E + E is called compact if it is continuous and maps bounded sets into relatively compact sets. If A is a bounded, selfadjoint operator on E the essential spectrum o,(A) of A consists of all elements of the spectrum of A which are not isolated eigenvalues of finite multiplicity. A functional I: E + R is said to be coercive if lim Z(z) = cc. I/~l/+m A functional I: E -+ R is called weakly continuous if it maps weakly convergent sequences into convergent sequences. Denoting weak convergence in E by - we call I weakly lower semicontinuous if zn - z in E implies I(z) < lim inf I(z,). THEOREM 3. Let E be a real Hilbert space, f E C2(E, R) such that Vf: E --f E is a compact mapping. A is assumed to be a bounded self-adjoint operator on E such that 0 4 a,(A). Let E, be the space AT/c I/: where A is positive definite and T/c E + a subspace satisfying I/ = V, I:E-Risdefinedby
Z(z) = +(Az, z) + f(z). We assume (a) (i) 11T/is coercive. (ii) -I is coercive on v’, uniformly on bounded subsets of T/ (i.e. for all M > 0 and R > 0 there exists 6 > 0 such that Z(u + w) < -R if u E r/; w E V’ and llzlll< M, llwll>, 6). (iii) For all z E E and all w E V’ \ (0) we have (Z”(z)w, w) < 0. Then c = inf sup Z(u + w) is a critical value of I. If in addition (1”(z)u, u) > 0 for all z E E and all vcv KC.V u E I/\(O) then 1’possesses exactly one critical point. (b) Let conditions (iHiii) be satisfied as well as (iv) 1’(O)= 0 and there is a nontrivial subspace t c T/such that (Z”(O)& u) < 0
for all nonzero elements in V + VI. Then I has at least one nontrivial critical point. (c) Assume (iHiv) as well as : (v) I”(O), interpreted as a bounded linear operator from E to E, has a bounded inverse i.e. r’(0) E GL(E). Then I has at least two nontrivial critical points. (d) Let condition2 (iHiv) be satisfied, I an even functional, Z(0) = 0. Assume that for p from (iv) we have dim V 3 k for a natural number k. Then I has at least k pairs of nontrivial critical points.
KLAUS
800
THEWS
Remarks (i) Theorem 4 of Castro and Lazer yields essentially the same conclusions but requires dim V < co. On the other hand Theorem 4 deals with a more general class of functionals than our Theorem 3. (ii) The existence statement in part (a) which is rather straightforward can also be derived from Berger’s Theorem (6.3.13) in [4]. Berger’s multiplicity statements however use Morse theory and are only generic, furthermore Vf is assumed to be uniformly bounded and A to have a finite Morse index (Compare with Theorem (6.5.16) in [4]). (iii) The assumptions on A and f and condition (ii) imply that I/ has finite codimension Proof: 0 4 a,(A) implies that there is a constant m > 0 such that
in E,.
~Az, z) 3 rnllz11’for all z E E,. We have E + = I/ @ I/’ where J’/’ = 1/l n E +, According to (ii) there is a 6 > 0 such that I(v’) d - 1 for u’ E V’, I(v’I( 2 6. Suppose that dim V’ = co. Then there exists a sequence (0;) in V’ satisfying v’ - 0 in V’ (and, of course, then 0: - 0 in E) and 11 u’,)I = 6. A s is well known the compactness of ?j” implies that f is weakly continuous (Theorem 8.2 in [9]) and therefore f(uh) + 0. For n sufficiently large we have Z(u:) = &4vb, vk) + f(ui) which is a contradiction
to our conclusion
3 $rn@ > 0
from condition
(ii).
Proof of Theorem 3. (a) Let g,: I/’ + R be the coercive functional defined by g,(w) = - I(u + w) = -+((Au, 0) - $(Aw, w) - f (u + w). As VI = v’ @ E, @ E_ where E_ is the space on which A is negative definite and dim ( V’ + E,) -c co we conclude that g, is weakly lower semicontinuous. (Compare with example 8.2 in [9].) Therefore there is an element F(u) E V’ satisfying g,@(o)) = min g,(w). WEV
Condition (iii) implies the uniqueness ForrEI/;w,wl,wzEI/‘wehave
of W(u):
<&(w,) - SXWZ), Wl
-
w2)
1 =
GI:‘(w,+
z.(w,
-
w,,)(wl
-
wz),
~1
-
wz>
dT
s 0
=
s1-
(I”(0 + wz + +“, -
w,,)b”l - ~21,~1
-
~2) dT
0
3
a(~, wl,
w2)IIw1- ~~11’ with a(~,wl, w2) > 0
if w1 # w2 according to (iii). This relation implies and is a strictly monotone operator. We claim that W: V + l/i is a C’ mapping.
that &: I/’ -+ (V’)* can only have one zero
A
reduction
method
for some nonlinear
Dirichlet
problems
801
Proof: W(v)is the uniquely determined element w in 1/l for which (Z’(u + w), @) = 0 holds for all 6~E T/l. Denoting by R the orthogonal projection 71:E + 1/l this is equivalent to 7to VZ(u+ w) =O, To solve the equation 7~oVZ(u + w) = 0 for w we use the implicit function theorem. Let u0 E y w,, E 1/l be given such that 71oVZ(u, + WJ = 0. We define the mapping F:Vx
l/l+l/lby(u,w)+noVZ(u+w)
which is a C’ mapping. We want to show D,F(u,, wO)E GL(1/l) and compute
= d/dt(n: oVZ(u,, w,, + tw,), w,>l,zO
= (I”(u, + WO)Wl,WJ = (7coZ”(V, + W&WI,w,) where wl, w2 E 1/‘-and of course Z”(uO+ wO)is regarded as an element of L(E). In view of Al/ c I/ and AV’ c 1/* we get 7t oZ”(U, + WJ = T + 7cof’@,
+ WJ
where T = A(V’: 1/l + 1/l. TE L(I/‘) is a Fredholm operator of index 0 and z of”(u, + WJ is a compact endomorphism of V as f”(u, + wO)E L(E) is compact as a derivative of a compact mapping (compare with Theorem 4.1 in [lo]). Therefore D,F(u,, WJ is Fredholm of index 0 so that it suffices to show: KernD,F(u,, wO)= (0). But this follows from (D,F(u,, wO)wI, wl) = (Z”(u,, wo)wl, w,) < 0 for w1 # 0 which is a consequence of (iii). So our claim follows from the implicit function theorem. After these preparations we are able to reduce the search for critical points of I to that of a simpler functional G. Definition. Let G: V -+
R be given by G(u) = Z(u + W(u)).
LEMMA 1. z E E is a critical point of Z if and only if z = u + W(v)and u is a critical point of G. Proof: Each critical point z of I has the form 2 = u + W(u).The assertion now follows from the formulae (G’(u), 6) = (d/dt)Z(u + tv” -t W(u + tv”))l,,o = (Z’(u + W(u)),5 + (W)yu)iQ =
where u, 6 E I/: The last equation implies that G E C2( K R) and :
(G"(u)u1,u2) = (I"@+ E(u,)(q + W'(u)u,), u2). We note that G is coercive as G(u) 2 Z(u) + 00. To prove the first part of (a) it is enough to show that G is weakly lower semicontin!%z ?m V because then G achieves its infimum on V which is a critical value of G and also of I. For preparation we prove:
KLAUS THEWS
802 LEMMA
2. The mapping
W: V + I/’ is bounded
on bounded
sets.
Proof. Let K > 0 be given. As I is bounded on bounded sets there is a constant L > 0 such that Z(u)3 -LforvEX I/u// < K.D ue t o condition (ii) there is a constant 6 > 0 with the property: For w E V1, u E V, llull < K, llwll 3 6 it follows I(u+w)d As Z(u + W(u)) = sup Z(u + w) we conclude wcv’ LEMMA 3. G:
-2L.
[Iw(u)[~< 6.
V + R is weakly lower semicontinuous.
Proof: Let rc’: E + V denote the orthogonal projection. For V,G which is the gradient with respect to the inner product on V induced by E we have:
of G
V,G(u) = 7~’o VZ(u + W(u)) = Au + rr’o Vf(u + W(u)), As G(u) = $4u,
v) + $(AW(u), W(u)) + f(u + W(u)) we conclude 71’0 Vf(u + W(u)) = V,($4w(u),
W(u)) + f(u + W(u))).
mapping and bounded The mapping u -+ rc’o V’(u + W(u)) is compact because W is a continuous on bounded sets and Vf is compact. Therefore the associated potential is weakly continuous. Now G is weakly lower semicontinuous as a sum of weakly continuous functional and a weakly lower semicontinuous one. The proof of the first part of assertion (a) is now complete. Concerning the second part, under the additional hypothesis, we remark that u + I(u + w) is a strictly convex function for fixed w E V1. The uniqueness result is then a consequence from Lemma 1 in [ 111. (b) Condition (iv) implies G’(0) = 0. It suffices to show that in 0 no local minimum achieved. For vi E V\(O) we have
for G is
(G”(O)o,, vi> = U”(O)(u, + W’(O)u,), vi> = (I”(O)(U, + w’(O)u,), u, + W’(O)u,) the last equation
being valid because (I’(u,
and all w E I/‘. Choosing
in particular
+ W(u,)), w) = 0 for all vi E V u1 E v\(O) we get
01 + (3ol)(a,) E (v” + VI)\(O) andlinally (G”(O)u,, vi) < 0. Therefore in 0 there is no local minimum of G and for the critical critical value c we have u,, # 0 because rc’u, # 0.
point
a0 belonging
to the
(c) The arguments here are based on two topological results the proof of which will be given in the next section. First we note that assumption (v) implies G”(0) E GL( V). Proof. An analoguous reasoning as in the case of D,F(u,, wO) shows that G”(0) is Fredholm index 0. So it suffices to prove that the kernel of G”(0) is trivial.
of
A reduction
method
for some nonlinear
Dirichlet
problems
803
Assume that v1 E V is such that 0 = (G’(O)v,, u) for all v E I/: This implies 0 = (Z”(O)(v, + W’(O)v,),U)for all u E I/ and in view of 0 = (I”(O)(v, + W’(O)v,),w) for all w E I/’ leads to 0 = Z”(O)(v, + W’(O)v,).Finally we get 0 = v1 + W’(O)v, and
0 = vl.
As shown above 0 and II,, = rr’r+,are two different zeroes of VG. We can assume that u0 is an isolated zero of VG and that VG has no zeroes the norms of which tend to infinity because otherwise(c) would be trivial. We now choose ((v,, II,)): = (Au,, vJ as a new inner product on 5 equivalent to the one induced by E and note that for the gradient of G with respect to ((.,.)), VG, we have ~G(v) = u + K(v) where K: V + V is a compact mapping. Now the assertion follows from properties of the Leray-Schauder degree. As G”(O)EGL( V) for sufficiently small I > 0 we have deg (VG, B,(O),0) = deg (@G)‘(O),B,(O),0) = -&1. From Theorem 5. in the next section we get for the local degree of VG at v,, which is a local minimum of G and an isolated critical point : deg(vG, B,(Q), 0) = 1 for small s > 0,
(*)
and for large M we obtain due to Theorem 4. deg (?G, B,(O), 0) = 1.
(**)
According to the relation deg (?G, B&J, 0) + deg (?G, B,(O), 0) # 1 ?G must have another zero in B,(O). This proves (c). (d) For the proof of this part we apply Clark’s Theorem 11 from [12] to our functional G. Before stating Theorem 11 we need a definition: A real valued Cl-functional f on a real Banach space E such that f(0) = 0 is said to satisfy (C) if every bounded sequence (x,) c E with the properties f(x,,) < 0, f(x,) bounded below and f’(x,) + 0 contains a convergent subsequence. The assertion of Clark’s Theorem 11 is : Let E be a real Banach space and f be an even, real valued, Cl-function on E. Suppose further that f satisfies condition (C), f(0) = 0, f is bounded below, f(x) 3 0 for large IIxI[,and f(x) = q(x) + o(IIx(12)as I/x(1-+ 0, wh ere 4 is a quadratic form of index k. Then f has at least k pairs of nontrivial critical points. We have to verify that G satisfies the assumptions above, As Z is even G is even and G(0) = 0. From the formula VG = id + K we conclude that G is bounded from below and that it satisfies condition (c):
KLAUS THEWS
804
Let (u,) c V be a bounded sequence such that VG(u,) --*0. As K is compact there is a subsequence u,,, such that K(c,,) converges, K(u,,) --) u in V: Since u,! + K(ti,,) 1_m + 0 we obtain u,, + - u. Lastly, from the hypothesis dim P > k and the expression for G”(0) we conclude that the dimension of space where G”(0) is negative definite is greater or equal k. The application of Clark’s theorem yields k pairs of critical points for G and therefore Z has at least k pairs of nontrivial critical points. Remark. Berger [4] calls a submanifold A4 of a Hilbert space H a natural isoperimetric constraint for a Cl-functional J on H if the following conditions are satisfied:
(i) c = inf J(u) is attained by an element f E M. UEM
(ii) For each ii E M n J-‘(c) if follows J’(U) = 0. (iii) Each critical point of J(u) lies on M. Using this notation it is easy to see that in our case where H = E and .Z = Z M : = {z E E, VZ(z) I V’} is a natural isoperimetric constraint for I. The described reduction method then consists of looking at Z only on M and using a parametrization for M: M = {(u + W(u)),u E V}, $:V+M u + (u + 5(u))
is a parametrization
3. THE
of M. Then for G we have the representation
LERAY-SCHAUDER
DEGREE
OF
SOME
SPECIAL
G = lo $.
GRADIENT
OPERATORS
This section is devoted to the proof of generalizations of two well known facts concerning the Brouwer degree of the gradients of some functionals, namely Theorem 4. which immediately implies formula (**) in the proof of Theorem 3. (c) and Theorem 5. which yields formula (“). THEOREM
4. Let H be a real Hilbert space and G E C’(H, R) be given by G(w) = $11~11”-f(w)>
where V’: H -+ H is a compact mapping. Furthermore G is
non-degenerate,
WGH
we assume that
i.e. there is a constant R > 0 such that
lim
G(w) = cc and that
liwli-+m
I[wI[>, R implies VG(w) # 0. Then, for sufficiently large M, we have deg (VG, BJO), 0) = 1. Remark. In the special case H = R” this theorem is just the assertion of Krasnosel’skii’s Lemma (6.5) in [13]. We make essential use of some ideas of Krasnosel’skii’s proof.
Next we formulate a local analogue of Theorem 4. :
805
A reduction method for some nonlinear Dirichlet problems THEOREM 5. Let H be a real Hilbert
space, y E 23, R an open neighborhood
of y and
G E C’(Q R) where G(w) = ill w(I2 - f(w) and V’: Q + His a compact Assume that y is a local minimum for G and an isolated Then, for sufticiently small r > 0, we have
mapping.
critical point.
deg (VG, B,(Y), 0) = 1. Remark. In the case G E C2(S& R), an assumption which is satisfied in our application, Rabinowitz, for instance, gives a proof of this assertion (see [ 141). But as his proof seems to be complicated we give another proof which is completely analogous to that of Theorem 4. Proof of Theorem 4. We begin with a preliminary LEMMA1. Let R > 0 be given. Then there is a separable Hilbert space V c H satisfying Vf (B,J c I/: Let (y,), E N be a complete orthonormal system for I/ and x,,: H + span {y,, . . . , y,} the orthogonal projection. Then for each E > 0 there is m(c) E N such that n 3 m(E) implies /IVf (z) - 71,o Vf (z)I/ d F
for all z E BR.
Proof. N: = Vf(B,) c H is compact and separable as {w,, n E N) be a countable dense subset of N. Then I/: = such that Vf (B,) c v Now assume that the second assertion is wrong. Then there are E > 0, sequences mj + 00 and zj E BR such IIvf(zj)
Using the compactness
-
n,jvf(zj)II
a compact metric space. Let M = . span M is a separable Hilbert space
that
3 ”
of Vf we can assume Vf (zj)j_,
-+ u E v.
Then ‘f =
v
+
('j)
-
%ljvf
(Vf(Zj) - u) - 71JVf(Zj)
('j)
- u) - 7&u
= (id - 7&J(u) + (Vf(Zj) - u) - 7c,,(Vf(Zj) - u). All terms on the right hand side converge
to 0 which leads to a contradiction
LEMMA 2. G satisfies
bounded
the Palais-Smale condition: Each sequence and VG(w,) + 0 contains a convergent subsequence.
to our assumption.
(w,) c H such that (G(w,)) is
Proof: (G(w,)) is . b ounded from above and therefore (w,,) is bounded taking into account the coercivity of G. Using the compactness of Vf we find a subsequence Vf (wnr) + w E H. Since Vf (w,,) we conclude wnk + w. As a consequence we note that for any closed VG(w,J = w,~ subset M c H on which VG has no zeroes there is a positive constant 6, > 0 such that IIVG(w)ll 2 6, for all w E M. After these preparations
we come to the actual proof: As G is nondegenerate
there is a constant
KLAUS THEWS
806
R, > 0 such that VG(w) # 0 for ljwli 3 M,:
R,.As G IS . b ounded on bounded sets we define =
sup G(w) < W. llwll
Using the coercivity of G we find that R,: = {w E H, G(w) d M, + 1) is bounded. Now we choose p > 0 and R, > p such that BR, _p I R, and define M, : = sup G(w) + 1 as well as I/WI,
let
< M,)
R2 begiven with the property Q2, = BRZ.
Since VG does not vanish
on {WEE&M, LQ: =
+ + < G(w) < M,f
Mo+ l;:nfilw,-_M, IIVGCw)II
is strictly greater than zero. Let K: = ,,w”pzR_tlVG(w)ll < co. 7 NOW we
apply Lemma
1, taking
R = R, and F = min{(a,/2),
En: = span (yl,. . . , y.} E, is equipped
with the inner product
and G,: E, + R induced
by
(c(i/4K)}. For n >, m(s) we define
by G,(w) = G(w)
for w E E,.
E.As usual, defining V,G,(u) by the requirement for all II E E,,
(V,G,(u), u) = (Gn(u), u)
we obtain V,G,(u) = rr, o VG(u) = u - n, o V’(u) for all u E E,.Define B,:
=E, n BR,. The essential step is now to compute the Brouwer degree of
V,G,. ASSERTION.
deg (V,G,, B,, 0) = 1 for n 2 m(c).
Proof: There is a locally Lipschitz
continuous
vector field H,: BRZ n
11H,(u) - rc, 0 VS(u)ll < F’ where E’: = min {a,/4, c$4H}. This implies the ordinary differential equation
E, -+E,satisfying
VuEE,ni?Rz
deg (II,, B,, 0) = deg (V, G,, B, 0). Now we consider
1 = -H,(x) (*)
x(0) = x0 in the domain
int (Q,)
n E,.Let x(t) be a solution
such that
M, + $ d G(x(t)) < M,.
A
reduction
method
for some nonlinear
Dirichlet
problems
Then we compute = -(VG(x(t)),
(dldr) G@(r)) = (VG(x(t)), i(r))
~,(x(t)))
< - a;/2
(**)
using the definitions of 01~)K, E and E’. Therefore, trajectories starting in int (0,) n E, remain there and the solutions are defined for all positive times. Let U(t) be the translation operator associated with (*) i.e. U(t)x is the solution of (*) with initial value x at the time t. All elements x E a!, c int(n,) n En are points of irreversibility for (*) i.e. for all t > 0 we have x # U(t)x, x E adl,. This follows because G is strictly decreasing along the solutions x(t) if x(t) E RI\&,. As H, has no zeroes on 8B, Lemma (6.1) in [13] gives the information: deg (H,, B,, 0) = deg (id Now relation
(**) implies that a trajectory ZE
and remains
starting
U(t), I?,, 0).
in 8B, is in R, at a certain
1
2&f, -of,+ 1))
6 o
there in future. Let T > 2(M,
2 MO
- (MO + l))/a: be a fixed number.
U(T)x E Cl0 n E, c B,, n E, The homotopy shows that
time
H(s, x) = x - sU(T)x,
O
deg (id -
Then
for all x E 8B,. XEB,
which
has no zeroes
on 8,
U(T), I?,, 0) = 1.
This proves the assertion above. Finally the definition of the Leray-Schauder degree, [15], taking X, : =E,, F, : = nn,OV'leads to the formula
as for instance
given in section
16 of
deg (id - f, BR,, 0) = deg (I,,, - 71,o Vf, B,,, 0)
which proves Theorem
4.
Proof of Theorem 5. Without loss of generality we assume y = 0, G(0) = 0 and y = 0 being a strict local minimum. We choose R, > 0 such that llwll G Roy We need a preliminary
w # 0
imply VG(w) # 0, G(w) > 0.
lemma:
LEMMA.For 0 < R, d R, we have inf
G(w) > 0.
R,QIIw~lQR Proof. Assume that the assertion is wrong. Then there are w,, E H, R, d llw,/I d R, satisfying G(w,,) -+ 0. As BRo is weakly sequentially compact we assume w, - w, w E BRI,. Consequently
11 w II < lim inf IIw, II and f(w,) + f(w).
KLAUS THEWS
808
From
G(w,) = tilwnII 2 - f(w,) + 0 we conclude G(w) = +l/wll’ -f(w)
411w,Il 2 + f(w). Therefore
d +lim IIw,(12 -f(w)
= 0.
This implies G(w) = 0 and w = 0. On the other hand, due to R, 6 IIwnll,we find llw,,1123 $RT > 0
fiw) 3 $liminf
and G(w) = G(0) d -iRf which is a contradiction. Now we come to the actual proof: Let0 < c(~: = inf G(w) be chosen. Define CI, : = {w E BRo, G(w) 6 a,/2} and choose 0~
R, <
II4=Ro R', such that BR, c BRi c n,. Then define cl, : =
inf G(w)>0 andQ,:= RI< II4 SRo deg (VG, B,,, 0) = 1 is proven completely analogous
4. PROOFS
as in the preceding
1 AND
OF THEOREMS
First we collect some standard facts concerning Let E: = WA, 2(G), f: E + R be defined as f(u) =
{w E BRo, G(w) < cr,/2}. Finally
P(x, u(x)) dx
the functional
where P(x, z) =
associated
for u E E. Due to our assumptions of (DP) and also classical
proof.
2
analytic
treatment
of (DP).
’ p(x, t) dt. s 0
sG The energy functional
the fact that
with (DP) is defined by
(i) and (ii) we have I E C2(E, R). Critical points of I are weak solutions solutions by known regularity results. For u, u E E
aijDiuDjv + c(x)m)
dx - jG p(x, u(x))v(x) dx holds.
For u, v, w E Ewe have pu(x, u(x))v(x)w(x) dx.
(Z”(u)u, w) = (u, w) sG As already mentioned elliptic operator L.
in Section 1. E will be equipped
yf: E ---fE f :E + R The linear eigenvalue
with the inner product
is compact
(. , .) arising from the
and
weakly continuous.
problem Lu(x) = Au(x), u(x) = 0,
xeG xedG
VA
A reduction
method
for some nonlinear
Dirichlet
809
problems
possesses an unbounded sequence of eigenvalues 0 < I, < I, < i3 . . . + co and a corresponding system (Q,) of eigenfunctions which is a complete orthonormal system in (E, (. , .)). For u E span {@I,. . ., @‘l1 we have the inequality
s s
u’(x) dx
(4 4 G 1, and for w E (span {aI,.
G
. . , rI$))’ we get
(w,w)3 Al+1 w’(x) dx. G
For proofs of these statements
and additional
references
Proof of Theorem 2. Our aim is the application We choose E = WA* *(G),
we refer to [6,16].
of Theorem
3 to find critical
points
of 1.
Z(u) = 411q2 - f(4 where
f(u) =
s
P(x, 4x)) dx
G
and
I/: = (span {Q,,, . . . , CD[})*, A: =id,. Then 1/l = span {aI,. Now we show that the assumptions Verification of(i). Using the right inequality f(u) =
of Theorem
in hypothesis
3. are satisfied.
(b) we obtain
P(x, u(x)) dx d $ sG
. . , ml}.
A,, 1u2(x) dx -
Z-Z,(x, u(x)) dx + const. s G
s
and therefore Z(u) > fllull’
- fA,+,
u’(x) dx + I’G
where the appearing constants v’ E E(A, + ,) and v” E (Et,$+ ,))‘.
H,(x, u(x)) dx - const. sG
are independent
of u. Let v E V be given,
Using that
11 v” 11’2
u”(x)~ dx
p
for a constant
sG P ’
A,+, and
that the system (a,) is also &-orthogonal Z(v) 2
(1 - %) IIv”l12
we conclude:
+ JGH2(x,v(x))dx
- const.
v = v’ + v” where
810
KLAUSTHEWS
As 1h,(x, u) 1is uniformly bounded and L’(G) continuously embedded in L*(G) it follows
/u”(x)1dx - const.
H,(x, u’(x)) dx - K
+
sG
sG
H,(x, u’(x)) dx - const.
+
sG The last inequality together with the hypothesis concerning H, yields the coercivity of I on I/. Verification of (ii). The left inequality in hypothesis (b) leads to I(u) d
($jGALu’(x) dx + jG H, (x, u(x)) dx -
+lluII2 -
cons,.).
Splitting
u=u+w,
v,
uE
WE
VI, w”
w’ E JW, ),
w = w’ + w“,
E
E(A, )’
we obtain:
44 G 3llull’ + lIw’l12 + IIw”l12) -
$A, G (u’(x)+ w’(x)~+ w”(x)‘) dx
-
s
s
H,(x, u(x)) dx + const.
G
u2(x)dx
< +Ilull” - +A, sG
w”(x)~dx
+ $llw”l12 - +A,
sG c -
J G
<
r H,(x, w’(x)) dx + K G Iv(x) + w”(x)Idx + const. J
fllull’ - $2,
s
u’(x) dx + K G 1u(x)] dx
G
+
-
w"(x)~
$tv -4)
s
s
dx + K G 1w”(x)1 dx
sG
H,(x, w’(x)) dx + const.
G
s
A reduction
method
for some nonlinear
Dirichlet
811
problems
Here Yis a constant satisfying v < ir. We have used that the eigenfunctions Oj are C-orthogonal and that (h,(x, u) ( d K. The assumption on H, and the last formula then imply condition (ii). Verification of (iii). For z E E, w E V”\(O) we get (Z”(X) w, w) = (w, w) -
s
JP,(x,z(x))w’(x)dx G
(A, - p,(x, 4x)))w’(x)dx < 0.
<
G
Now we have shown that assumptions (a’) and (b) of Theorem Theorem 3. This proves Theorem 2, or). Next we show that condition
(iv) is a consequence
Verification of (iv) p(x, 0) = 0 implies Z’(0) = 0. Choose
2, imply conditions
of the first hypothesis
(iHiii) from
in (/I).
v : = R. (I$+ 1. Then for u E p + 1/l we have
(Z”(O) u, u) = (u, u) -
pu(x, 0) u2(x) dx sG
d
((4,
J
-
P,,(x,
0))
u’(x)
dx
<
0,
s G
ifu # 0. Part (b) of Theorem additional assumption
3 now proves the first assertion in (j?) implies hypothesis (v).
of@). The second assertion
is true since the
Verification of(v). Z”(0) E L(E) is Fredholm of index 0. Therefore it is enough to show that Kern Z”(0) = that there is u E E such that Z”(0) u = 0. Then (Z”(0) v, u) = 0 for all u E E and (u, u) u(x) u(x) dx = 0 for all u E E which means that u is a weak solution of
01. Assume G pU(x, 0) x
Lw = pU(x, 0) w, G aG.
w =o, But then the additional
hypothesis
shows that v = 0.
Part (c) of Theorem 3 then yields the second assertion of@). Finally, to prove part (y), we remark that, under the assumptions of(y), Z(0) = 0 and Z is even since P(x, z) is even in z. For t we choose span {QL+r, . . . , QD, +,} and notice that for u E v + I/’ = span{@,, . . . , @,I+ k} the following relation holds: (Z”(0)
u, u)
= (u, u) -
p,(x,
0) u2(x) dx
sG -
ifu # 0.
p.(x,
0)) u”(x)
dx < 0
KLAUS THEWS
812
Part (y) now is a consequence of assertion (d) in Theorem 3. This completes the proof of Theorem 2. Finally we indicate how Theorem 3 can be used to prove Theorem 1. As the details are similar to the proof above we shall be sketchy here. Proof of Theorem 1. This time, with the aid of Theorem 3, we are looking for critical points of -I on E = w:,‘(G). We define V = span{QI, . . . , (I+} and the operator A is defined by Au: = v - w
where for
UGE
we set u = v + w, v E V, w E 1/l. Then
44 =
=
-(illull’ - JG p(x,U(XW) -+llwll’ + +I~~’ +
(j-p(x, u(.y))dx- Ilull’). G
We shall briefly mention how the various assumptions of Theorem 3 can be verified in this situation. Conditions (i) and (ii) follow from hypothesis (b) using similar arguments as before. Condition (iii). For z E E, w E V”\(O) it follows: (I”(z) w, w) = -(w, w) +
s G
s
PAX,z(x))w’(x)dx
Q G(PAX, 4~)) -
4 +,) w2(x) dx < 0
in view of hypothesis (a). This proves the first assertion of part (a). Quite similar reasoning shows that the additional hypothesis in (a) implies that (Z”(z) v, v) > 0
for all
z E E, v E V\(O).
From this the uniqueness of the solution can be deduced. Condition (iv). This is satisfied in view of (c) with t:=
R.Q,.
Condition (v). This assumption is a consequence of the additional hypothesis in (p). Now (8) is a consequence of parts (b) and (c) of Theorem 3. Condition (vi). Choosing I/: = span{@,_,+ 1, _. . , CDI ) (vi) is satisfied under the assumptions of (y). Assertion (y) is then a consequence of part (d) from Theorem 3. This completes the proof of Theorem 1.
A reduction
method
for some nonlinear
Dirichlet
problems
813
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