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Generalization of Fredholm alternative for nonlinear differential operators
Nonhecv Ana&~ir. Tkeory. Merkodr & Apphzriom. Pnnred 1” Great Einraln.
Vol
10. No.
10. pp. 1083-1103.
19%.
0362-5%X,‘86 P3.M) - .M) Perqamon Journals Ltd.
GENERALIZATION OF FREDHOLM ALTERNATIVE NONLINEAR DIFFERENTIAL OPERATORS
Department
FOR
Lucre BOCCARDO Universita di Roma “La Sapienza”, P. A. Moro 2, 00185 Roma. Italy
of Mathematics,
PAVEL DRABEK Department
of Mathematics, Technical University of Plzeti, Nejedleho sady 14, 30614 Plzeti, Czechoslovakia DANIELA
Department
of Mathematics,
GIACHETTI
II Universita di Roma, Via Orazio Raimondo. 00173 Roma. Italy
and MILAN
KUCERA
Mathematical Institute of Czechoslovak Academy of Sciences, iitna 25, 11567 Praha 1, Czechoslovakia (Received for publication 21 October 1985) Key words and phrases: Nonlinear noncoercive equations, nonlinear eigenvalues, Fredholm alternative
1. INTRODUCTION WE SHALL
consider the boundary value problem
-(Iu’(t)lP-* u’(t))’ =f(t, u(t)) + g(r), t E (0, ‘2). u(0) = u(n) = 0.
I
It will be proved that if
lie (roughly speaking) between eigenvalues of the eigenvalue problem - (Iu’IP-~u’)’ = AIulP-*u
on
(0,x) (EP)
u(0) = U(ir) = 0
1
then (BP) is solvable for anyg E L, (theorem 2.1). Moreover, certain more general assumptions about f under which (BP) is solvable will be formulated using properties of the problem - (Iu’IP-‘u’)’ = ,uIuIP-~u+ - ~luIp-~u_
on(0, x) (EP,, y)
u(0) = u(n) = 0
1
(theorem 2.2). Further, we shall prove under the similar assumptions about f the solvability of the equations asymptotically close to (BP) (theorem 2.3). Certain weaker results will be obtained for partial differential equations of the analogous type (theorems 2.4-2.6). Let us remark that for the special case of differential operators under 1083
1084
L. BOCCARDO et al.
the consideration our results represent generalizations of abstract Fredholm alternative for nonlinear operators proved by NeEas [19], Pochoiajev [20] and developed by many authors (see [12] for further references). Much work has been devoted to the study of differential or abstract nonresonant semilinear equations of the form Lu + Nu = h with L linear and N nonlinear operator In the differential case
(see e.g. [lo, 2, 171).
LJA= - ,i;.F<, (-l)“‘D’($j(x)D’u)
=f(x, n),
existence and uniqueness results are given by Dolph [7], in the framework of integral equations and by Landesman and Lazer [lj], Lazer and Leach (161, Kannan and Locker [13]. Other results are given by Williams [22], de Figueiredo [6], Kazdan and Warner [14], Thews [21], Castro [3], Castro and Lazer [4], Ambrosetti and Mancini [l], provided there exist real numbers p, q such that A
11) q < l.,t,.
--s
I
U
The general case fb. pi ~ p(.r) ~ 11
u>s q(x)
< A,_,
has been studied by Mawhin and Ward [18]. However Lazer and Leach showed in [16] that the previous inequalities cannot be replaced by
A.
11
‘171.
In our problem the principal part also is nonlinear. generalization of the previous ones.
Therefore,
our results are a partial
Notarion. Let p 2 2. We shall denote by I@; the usual Sobolev space on (0, X) or on Q (the domain in Rv), Wp;’ its dual space, (. , .) the pairing between I8’; and Wp’_?.The norm in I@. and in W;l will be denoted by I].// and ]I.jj*, respectively. The symbol B: will denote the ball in Wp’ with the radius r centred at the origin, deg [T: B: , 0] will be the Leray-Schauder degree of the mapping T: Wp’_?---f IV;’ at the origin with respect to B:. The strong convergence and the weak convergence will be denoted by + and -, respectively. We shall denote by & (i = 1,2, . . .) the eigenvalues of (EP). It is known that the eigenvalues of (EP) form an increasing sequence {A,}of positive numbers, An-, + r: (see [12, 81 and remark 3.2). By the solution of a boundary value problem for ODE we shall always mean a function u E C’((0, x)) satisfying the corresponding boundary conditions such that Iu’~J’-‘u’ is absolutely continuous on (0, n) and the corresponding equation holds a.e. in (0, n). By the solution of an initial value probfem on (to, T) we mean a function u E C’((to, T)) satisfying the initial conditions such that ]u’IP-%’ is absolutely continuous on compact subintervals of (r~, T) and satisfying the corresponding equation a.e. on (to, T).
Generalization
1085
of Fredholm alternative
The positive and negative part of u will be denoted by U- and u-, respectively. The function f is supposed to satisfy Caratheodory’s conditions automatically. 2. FORMULATION
OF IMAIN RESULTS
AND SOME
COMMENTS
A. Ordinary
(Eo) ’ S_+r IsIp-Zs c “~S~P
-IsIp-zs
s *i+t
- 6
for a.e. t E (0, n) with some 6 > 0 and i 2 1, where A, (n = 1,2. . . .) are the eigenvalues of (EP). Then (BP) has for each g E Lr at least one solution. Remark 2.1. Let K be the square from Fig. 1 and Fig. 2, respectively. respectively, can be written as
litn~;p#$)-,I__~p~] S
E K,
S
Then (E,) and (E,),
fora.e.tE(O,n) e>
lim inff(t' s_+* IsIP-ts'
liminf I__I
f(tJ' IsIp-2s
E I
K
'
fora.e.tE
(O,j-o.
Further we shall describe more general set K such that (E) guarantees (BP).
Fig. 1
the solvability of
L. BOCCARDOer al.
1086
Fig. 2.
Remark 2.2. Denote
by A-, the set of all couples [p, V] E IL!*such that the problem (EP,,.) has a nontrivial solution. It is proved in [8] that [II, V] E A_, if and only if one of the following conditions is satisfied (see also remark 3.5): (i) p = Al, v arbitrary (A,, are eigenvalues of (EP)); (ii) p arbitrary, v = Al; (iii) p > Al, v > Al and at least one of the numbers w,(p, v), w,(p, v), W&L, u) is positive integer, where VP Wl(lA
v)
=
Wj(,U,
v)
=
(pl,pk;
. v’lP V”P)(A,)“P
(vVP
-
~yP)y’iP
VP
+
v UP
( ,U
Fig. 3.
)A
’
1’”
’
Generalization of Fredholm alternative
1087
The set consists of the lines intersecting the axis ,U= Y at [I.*,,I,] (i = 1,2,. . .), see Fig. 3. Put A, = IW\A_,. It is proved in [8] that if [,u, V] belongs to any component of A0 containing some [j_, A] (in this case it is A # Ai, i = 1,2,3, . . .; see Fig. 3) then for any g E L1 there exists a solution of the boundary value problem = ~~KIP-*K+ - ujc~lP-*~- + g
-(IK’IP-*K’)’ U(0) =
427)
=
on (0, x), (BPU.“)
0
(cf. remark 4.2). (Particularly, for any ,UE R, even if p = A,, there are v such that (BP,..) is solvable with any g E L’). This can be understood as a generalization of the Fredholm alternative for the problem (BP) withf(t, U) = $@-*u (for Fredholm alternative for nonlinear equations see e.g. [12])*. The following theorem 2.2 is a generalization of this existence result as well as of theorem 2.1. THEOREM 2.2. Suppose that f satisfies (G) and that there exists a component AL of A0 containing a point of the type [A, A] and points [,u,, vi], [lu,il, vicl] E Ai such that
for a.e. t E (0, n). Then (BP) has for any g E L, at least one solution. Remark 2.3. The index i has a precise sense in theorem 2.2. if we denote by Ai andAd (i = 1,2,. . .) the component of A,, containing a point [I, A] with A < At and ,I E (&, I.;+,),
respectively (see Fig. 3). In the case i = 0 the estimate from below in (EA) is not necessary (i.e. we can formally put y, = v. = - x); this case is the same as (E,) in theorem 2.1. The condition (E/) for i 2 1 is a generalization of (Ei) which corresponds to the case !ci = I/i = pi + 6, pi+1 = Vi+, = I-i+1 - 6. The condition (El) is equivalent to (I?) from remark 2.1 with the oblong K = {[v, nz] E W; /J E (,Ui,,~i+l), v E (Vi, Vi+r)} (see Fig. 3, cf. remark 2.1). Remark 2.4. Let J, F: I@; + W,! (J(u),
be operators
such that
u) = _/x ju’JP-2u’u’dt,
for all
K,
u E
I&’
b,
0
(F(u),
u) = \“f(r.
u(t))u(t)
dt,
for all
K, u E @ L.
0
The assumption (G) guarantees that F is well defined. (The proofs in Section 4 nork also with q = 1 in (G) if we use the fact that bounded equi-integrable subsets of L1 are weakly * Let us realize that (BP,..) can be written as (BP) withf(t, u) = ,~lulp-~u, u > O,f(t. u) = vIuIp-‘u, u C 0.
L. BOCCARDO~~~~.
compact instead associate g’ E W,-? defined by
L,. q > 1.) To an arbitrary g E L1 we shall
the reflexivity
(g*, u) = 1” g(t)u(r) dt,
l&k+;.
for all
0
If u E k!‘i is a solution of the abstract equation J(u) - F(U) = g*,
(AE)
then u is the solution of (BP) (in the sense mentioned in Notation, Section 1). This can be shown using the standard regularity argument for ordinary differential equations (see e.g. [8]). Remark 2.5. If f(t, U) = A]u]P-*u, where A is not an eigenvalue
of (EP) then the assertion of theorem 2.1 follows from abstract Fredholm alternative for nonlinear operators (see e.g. [12, 19, 201). Following the ideas from [12, 191 we shall generalize theorems 2.1, 2.2 for the equations asymptotically close to (AE).
Definition 2.1.
Let A0 : I@; + W>’ b e an a-homogeneous operator (a > l), i.e. Ao(tu) = uE I@;. Then A: I!‘;W;! is said to be a-quasihomogeneous with
t”AO(u), for all t>O, respect to A0 if
t,-+O,u,-u,t;A
-+g*
in
WP;‘,
imply g” = A,(u). The operator A is said to be an a-homeomorphism
of I!‘; onto Wpl if it is a homeomorphism
of fi; onto Wp;’ and there exist Ci, C2 > 0 such that C,IIuII” s ItAut]*s C21141'
(2.1)
for all u E tii. Further, we shall suppose that there exist
. f(t9s) ,!F%IsIp-2s and consider the estimates
,l&lx
f(t, s>
5s Al
IsIp-2s
f(tt s>
- 6, ,“y
szAl - 6;
lslp_*s
f(4 s)
f(t73)
Ai + 6 6 s_+x lim -~~~~~~~ s ki+i - 67 Ai f(t,
pi
a.e. in (0, Id).
G
s)
W-o)
+
6
s
,gyz
f(4
IsIp-*s
s>
s
Ai+l
vi G I__x lim -IsIP-*s -= Vi+‘,
-
6;
W-J
(EL,‘)
1089
Generalization of Fredholm alternative
2.3. Consider the situation from theorem 2.1 and 2.2 but replace (E,), (E,) and (El) by (H-c,), (EL,) and (EL!), respectively. Let A be an odd (p - 1)-homeomorphism of I@; onto W>’ which is (p - l)-quasihomogeneous with respect to J. Then for any g* E Wp;’ there exists at least one solution of THEOREM
A(u) - F(u) = g*.
(A%)
that q =p* (for q > 1 see condition (El). Then the assertion of theorem 2.3 remains true.
THEOREM 2.4. Let us suppose
(G)),
(E,),
(Ei) and
Remark 2.6. The proofs of theorems
2.1-2.4 (see Section 4) are based on the properties of the Leray-Schauder degree and on some results concerning the corresponding homogeneous equations formulated and proved in Section 3. The proofs of these results in the cases (E,), (E; ) or (EL,), (EL; ) for i 2 1 are based on the method which cannot be used for partial differential equations. Therefore the results formulated further for PDE are weaker than previous ones for ODE. B. Partial differential equations
In this part we shall consider a bounded domain R C RN with a Lipschitzian boundary JR, the boundary value problem
u=OonaS2,
and the eigenvalue problem
=0 Remark 2.7. Let us define operators
J, S:
on lf’;-W,;
1 an. by
(J(u), v) =
forallu,vEW~andletF:W~+W;!,g*EW; ? be defined in the same way as in remark 2.4 but by means of integrals over 52. The growth condition (G) will be replaced by jf(X, $1 Z m(x) + cjs/p-‘, for all s E R, a.e. x E Q, with some mE
L,.
(i+$=
l),c>O,
(G’)
L.Bocca~oer al.
1090
which guarantees that F is well defined and continuous. (A more general assumption would be sufficient for the continuity of F, see [ll]; but we shall need (G’) in the proofs in Section 4.) The operators F, S are completely continuous with respect to the completely continuous imbedding fik C L,. A function u E fij is said to be a weak solution of (B?) if it satisfies J(u) - F(u) = g*. Analogously,
the weak solution of (fi)
(AE)
for a given J. is the function u satisfying
J(U) = AS(u). A number J. is said to be an eigenuafue of (ET) if there exists the corresponding nontrivial weak solution (the eigenfunction) of (l?P). It is well known that the Ljusternik-ScJhnirelmann theory ensures the existence of an infinite sequence of positive eigenvalues of (EP), but this theory does not give all eigenvalues in general (see e.g. [12]). The set of all eigenvalues can be more complicated and we cannot speak about the ith eigenvalue in the following theorems because we must consider all eigenvalues, not only those obtained by Ljusternik-Schnirelmann theory. However, there exists the least eigenvalue Ai > 0. THEOREM
2.5. Suppose that f satisfies (G’) and one of the following estimates: li,m_~~p$~*, i. - ci + 6 5
-6
S
f(X>s)
lim J-+Ix lsp-2s’
lim
f0ra.e.
xER;
0%)
sup f(X> s) S_.zr jsp-2s S A + ck - 6
(E).)
for a.e. x E Q; with some A which is not an eigenvalue Then for any
of (l?P), 6 > 0, where C~ = ,_$i, Mu) -
m4ll*
there exists at least one weak solution of (KP). Remark 2.8. It is easy to see that CA> 0 if 13.is not an eigenvalue
continuous
because S is completely
(remark 2.7).
2.6. Let A be an odd (p - l)-homeomorphism of t&‘i onto W;’ which is (p - l)quasihomogeneous with respect to J. Suppose that f satisfies (G’), (E,) and (E,). Then for each g’ E WP;’ there exists at least one solution of
THEOREM
A(u) - F(u) = g*. Remark 2.9. The case (E,) in theorems
(A%)
2.5, 2.6 is precisely the same as in theorem 2.1-2.4 for ODE because the corresponding information about the homogeneous equation (theorem 3.3) can be obtained using the variational characterization of the first eigenvalue. This is not
Generalization
of Fredholm
1091
alternative
possible in the case (Ei), (EL,), for i 2 1, p > 2 (cf. remark 2.6) and therefore the validity of an analog of theorems 2.1-2.4 for the case of PDE seems to be an open problem. In the case p = 2 variational approach can be used even in the case i 2 1 (see remark 3.6) and therefore the following theorem holds. 2.7. If p = 2 then the assertion of theorem 2.1 and the assertions of theoremsz.3, 2.4 in the case of the situation of theorem 2.1 holds with (BP), (EP) replaced by (KP), (EP).
THEOREM
Remark 2.10. We cannot speak about an analog of theorem 2.2 or of the second case in theorems 2.3, 2.4 (i.e. assuming (E/ ), (ELI )) b ecause vve have no corresponding information about the problem (EP,_) for the partial differential case. Remark 2.11. We are going now to explain a different procedure in order to prove the existence of solutions of the problem (BP), under the hypotheses of theorem 2.2. The approach is quite different from the previous one since no argument from degree theory is used and the solution is found by approximation. ru’evertheless, both the proofs use the “shooting method” (see theorem 3.1) as an essential tool. Let us only sketch the main steps of the proof. (i) We define U, as a solution of the equation
-(IU’(t)p-V(t))’
=fn(f, u(t)) + g(t).
t E (0, ;r)
u(0) = u(x) = 0, where fn(t, s) = S~uil~IJ’-‘~ + ~$,(t, s),
and T,, is the truncation operator at n. The existence of such a solution is guaranteed by a theorem of [12]. (ii) The sequence (~,),,~~v defined in (i) is bounded in fij. The proof is obtained by contradiction, using the “shooting method”. (iii) The sequence (~,),,~.v is compact in I&k. The cluster points are solutions of the equation (BP). 3. INVESTIGATION
OF THE
CORRESPONDING
HOMOGENEOUS
EQU,\TIOXS
In this section we shall prove some results which will be the bases for the proof of theorems 2.1-2.7. A. ODE of the second order with constant coeficients First of all, we shall consider the special case (theorem 3.1) which is sufficient for the proof of theorem 2.1 and only then we shall formulate more general theorem 3.2 which will be used for the proof of theorems 2.2, 2.3, 2.4.
L. BOCCARDO er al.
1092 THEOREM
3.1. Let x E L i satisfy one of the following estimates: x(t) 5 A1 - 6
a.e. on
Ai + 6~X(t)~~i+l
- 6
(0, j-r),
a.e.on
(Et)
(0,X),
(E?)
with some 6 > 0, where A, are the eigenvalues of (EP). Then there is no nontrivial solution of -+4-2w’)’
= xIwIP-Q$J
on
(0,~)
W(0) = W(Jr) = 0.
(EP,) 1
THEOREM 3.2. Let x+, x- E L, and let there exist a component Ai (for the notation see remarks 2.2, 2.3) of A0 containing a point of the type [A, A] and points [,Ui,vi], vi+ ,] E AL such that [Pi+l9 pi S x+(t) S pitI,
Vi 5 x_(t) s vi+1
a.e. on
(E;=)
(0, n).
Then there is no nontrivial solution of the problem -(Iw’IP-~w’)’
= ~+IwIp-~w*
- ~_IwIp-~w-
on
(0,x),
W’, )
w(0) = w(n) = 0.
1
Remark 3.1. If A& are defined in remark 2.3 then in the case (Ei’)
the estimate from below is not necessary, i.e. we can set formally p0 = v0 = - = (cf. remark 2.2). B. PDE THEOREM 3.3. Let x E Z+JS2) satisfy (Ei), there is no nontrivial weak solution of
where A, is the least eigenvalue
of (E?P). Then
THEOREM 3.4. (Linear case.) If p = 2 then the assertion of theorem 3.1 holds with (EP,) and (EP) replaced by (EP,) and (EP), respectively. Remark 3.2. The proof of theorem 3.1 in the case (Ei) is based on the variational characterization of A, and this method can be used also for PDE (theorem 3.3). An analogous approach for i 2 1 can be used only in the linear case p = 2 and only for this case we are able to prove the complete analog of theorem 3.1 for PDE (theorem 3.4), cf. remark 3.6. But in the case (Ef), i I 1, p > 2 the proof of theorem 3.1 will be based on the comparison lemma 3.1 and this will be proved by shooting method which cannot be used for PDE.
1093
Generalization of Fredholm alternative
Remark 3.3. For the proof of theorems 3.1 and 3.2 we shall use the properties
of the following
two initial value problems -(Iq-*~‘)’
= +JP-*u,
ll(t0) = 0, u’(to) =
(Y,
I
UPi)
and -(Iw’I@w’)’
=
xlwlp-‘w,)
WJ
1
w(hl>= 0, w’(h)) = p,
respectively, where to, A E R and x E L,(to, T) for some T > to, x(t) Z r z=-0 a.e. on (to, 7). The existence of the solution can be proved by standard methods (see e.g. [j]) and it is not hard to show that the solution is uniquely determined [8]. Further, it follows directly from the equation (IP,) that the solution is concave and convex on the intervals where w > 0 and w < 0, respectively. Particularly, this is true for (IP,) with A > 0. LEMMA 3.1. Let ui and u2 be the solutions* of (IP,,)
and (IP,,),
respectively.
with some to,
A,, AZ, (YE R, A,, AZ> 0. Set t& = inf{t > r,; u;(t) = 0). Suppose that x E Li(to, r), t,~ (to, T), where
w is the solution cf (IP,) on (fo, T) with /3 E W, (u/3> 0, and t, = inf{t E (to, T); w(t) = 0).
If Ai ~5x(t) s A2 a.e. on (to, T) then fl, Z t, ZGtA,. Proof. We can suppose (Y> 0, /3 > 0 without loss of generality.
Let us prove fz E ti,. We shall write A and u instead of A2 and 11~for the brevity. There are uniquely determined points ri E (to, tl) t; E (to, fx) such that u’(ti) = 0, w’(t;) = 0, u’(t) < 0 w’(f) < 0
on
on
(tf, tA), (3.1)
(tk, fX)
(see remark 3.3). The function G(t) = u(t - t; + ti) is the solution of -(]$(p-2$)’
= j+(P-Q, (R)
Lz(t, + t; - t;) = 0, fi’(r, + t; - ti) = CC. We shall prove that ti.+t’-tl,st x
X’
(3.2)
* In the sense mentioned in Notation (Section 1); remark that U,are defined on (r,,, + =) and that rL,are finite numbers, precisely see remark 3.4.
L. BOCCARDO er al.
109-I
Suppose the contrary (see Fig. 4). It is (w/il)(t;) > 0, (w/ti)(t,) = 0 and therefore there exists TV (t;, tx) such that (w/c?)‘(t) < 0, i.e. (w’U - WC’)(~) < 0, i.e. also F(t) CO”, where F(t) = (~w’]p-%‘(ti)p-’
- (W)P-i]ti’IP-%‘)(t).
Simultaneously, F(t;) = 0 because w’(t;) = 0 = ti’(r;) and therefore A C (t;, t) of positive measure such that F(t) < 0, F’(t) < 0
there exists a set
on A.
(3.3)
We can write F’(t) = F,(t) + F2(f). where F,(t) = (iw’lp-‘w’)‘(ti)p-’
- (W)“-‘(Iil’I~-~ti’)‘](t).
F?(t) = (p - l)(w’~‘(Iw’jp-~up-?
-
iti’lp-2~~~-2))(f).
It follows from (3.3)* that (w’ic - WC’)(t) < 0 on A and this together with (3.1) gives fEA.
F?(f) > 0, Further.
(3.4)
(IP,), (IP,) imply F,(t) = (A - ~(t))W(t)%i(t)~-l
2 0
a.e. on (0, X)
(3.5)
and (3.4). (3.5) contradicts to (3.3). Hence (3.2) is proved. Now by analogous considerations as above but on the interval (to + t; - ti, r;) instead of (t;, tA + t; - ti) we can prove t, < to + t; - r; and this together with (3.2) implies f, 2 t;, = fA,. The inequality fX 5 tA, can be proved by the analogous considerations. Remark 3.4. If uI is the solution of (IPJ with (Y> 0, A > 0 then u:, is 2(A,/A)“P ,r-periodic function having isolated zero points
rf=fo+j
A, T
i
VP
)
n,(j=0,1,2
)... ),
and lu~(r~)i = N. It follows from elementary investigation of (IP,) (cf. [S]). We can norm all eigenfunctions of (EP) in such a way that u’(O) = 1 because u’(0) # 0 for any nontrivial solution of (IP,) with respect to the unicity of the solutions (remark 3.3). Then A is an eigenvalue of (EP) with the corresponding eigenfunction uE.if any only if uj. is the solution of (IP,) with to = 0, LY= 1 satisfying u*(X) = 0. * We
use
the fact that C(r), w(r) > 0 on (t;, rx) and the function z--* IzIp-*z is increasing.
1095
Generalization of Fredholm alternative
follows from here that all eigenvalues of (EP) are of the form Ai = iP A, (i = 1, 2, . . .) that tfl = I-C.If A c Ai and A > Ai then tt > n and t” c x, respectively, where tf denotes the ith zero point of the solution of (IP,) with t, = 0, LY> 0. It and
Proof of theorem 3.1. Consider the case (Ei). If w is a nontrivial solution of (EP,) then we obtain (multiplying (EP,) by w and integrating by parts)
IK
Iw’IP dt
1=5
I0
.
(3.6)
~1wIP dt
It is well known that
(3.7)
and this is the contradiction because (3.6), (3.7) cannot hold simultaneously under the assumption (E;). Consider the case (El), i 2 1. Let Di and Vi+1 be the solution of (IPA) with i = i.i + 6 and A = /I,+1 - 6, respectively, and with (Y= 1. Denote by . ‘;=I
A,
VP
( 1 A,+6
n
t’+l 91
-j
(Ai+l1-
J”
n
o’= 1,2,. . .) the zero points of Uiand u.r+l, respectively (see remark 3.4). If w is a nontrivial solution of (EP,) then we can suppose w’(0) > 0 with respect to the unicity of the solution of (IP,). Hence w is the solution of (IP,) with to = 0, p > 0 and it is easy to see (using lemma 3.1) that t~+‘Stj+Stj
,...,
i),
where t,? denotes the jth (from the left) zero point of w (see remark 3.4). (It follows directly for j = 1; further, we can use lemma 3.1 at to = t; successively for j = 1, 2, . . . , i - 1 for the comparison of zero points of ui(t) = Ui(t - tr + $), W, Ui+,(t) = Ui+l(t - t,?’ + $‘I).) NOW, it follows from lemma 3.1 that w has no further zero point in (tr, n) because tiz] > ,7 (remark 3.4) and therefore w(x) # 0 which contradicts to the boundary conditions in (EP,). Remark 3.5. For the proof of theorem 3.2 we shall consider the initial value problem -(]4P--zu’)’
= pluIP-*u+
-
vIuIP-~u-, 1,
u(0) = O,u'(O) = y
Pu.“1
J
with y, Y, y E R, y # 0. Using remark 3.4 we can realize that u is the solution of (IP,.,) with y # 0 if and only if u is the solution of (IP,) with A = ,u, (Y= I;/( and with A = V, rr = -jyI on
1096
L. BOCCARDOer al.
any interval (to, T) where u is positive and negative, respectively. unicity of the solution of (IP,.,) and therefore nontrivial solutions Any nontrivial solution can be multiplied by positive constant such -1 and therefore we can consider the cases y = 1, y = -1 only. It that the solution u of (IP,,,) is periodic with the period (C)l”
+ (2)‘lP)
Particularly we have the can exist for y f 0 only. that u’(0) = 1 or u’(O) = follows from remark 3.4
n.
The solution UT” of (IP,U,,) with y = 1 has zero points f$” given by
The zero points rf:” of the solution uF” of (IP,U,,) with y = - 1 are given by analogous formulas but the role of ,u, Y in the right-hand side is interchanged. If ~1, v are given then a nontrivial solution of (EP,. “) exists if and only if UC” (‘2) = 0 or u?“(n) = 0 and this is fulfilled precisely for [,u, V] E A-, (see remark 2.2). More precisely, it is t,y;” = nor r;:” = n for [p, V] E A’_,, where Ai, is the component of A_1 containing [Ai, Ai] (Fig. 3). If [,u, V] E Ai (see remarks 2.2, 2.3) then t,r;” < X, ly:” < n, ttJi,+ > .z, f$Y,,- > n. Proof of theorem 3.2. Suppose that w is a nontrivial solution of (EPA=). Let u, and ui+, be the solutions of (IP,.,) with !L = yi, u = vi and ,~l= pi+i, v = vi+,, respectively, and with y = sign w’(O) (it is w’(O) # 0 with respect to the uniqueness). Denote by rj and ff*’ (j = 0, 1,2,. . .) the jth (from the left) zero point of ui and ui+i, respectively. The function w is the solution of (IP,) with x = x1 and x = x- on any interval where w > 0 and w < 0, respectively. It is easy to see from lemma 3.1 and remark 3.5 that
t~+‘5f,~5rj
,...,
i),
(3.8)
where tr is the jth (from the left) zero point of w. (It follows directly from lemma 3.1 for j = 1; further, we can use lemma 3.1 at to = ty successively with j = 1,2,. . . , i - 1 for the comparison of the 0’ + 1)st zero points of u,(t) = Ui(t - ty + ti), W, uitl = u,_,(f - t,? + ti”).) Now, using lemma 3.1 at tr once more, we obtain that w has no zero point in (tr, n) because t;:f > n (see remark 3.5). Hence, w(n) # 0 which is the contradiction. Proof of theorem 3.3 is the same as the first part of the proof of theorem 3.1 (the case (E,)). Remark 3.6. On the basis of the variational characterization of LS-eigenvalues (i.e. eigenvalues which can be obtained by Ljusternik-Schnirelmann theory, see e.g. [12]) it is possible to prove the following.
1097
Generalizationof Fredholmalternative COMPARISONLEMMA.If AAand A: is the nth LS-eigenvalue
of the problem
-~&(~~~p-2~) =AxIuI~-~u 1
I
I
u=O
onS2, 0naR
(*) I
with x = xl and x = x2, respectively, and x1, x2 E Lx(Q), x, s x2 a.e. in Q, then On the basis of this comparison lemma the following assertion can be proved. PROPOSITION.If x E L,, pi S x (x) C picl a.e. on R, where yi, pi+l are such that there is no eigenvalue of (I?P) in (pi, pi+l), then A = 1 is not a LS-eigenvalue of (*). But we cannot use this assertion instead of theorem 3.1 in the case p > 2 because it does not e?clude the possibility that A = 1 is (not LS) eigenvalue of (*), i.e. the nontrivial solution of (EP) can exist. In the case p = 2 there are only LS eigenvalues and therefore the mentioned proposition is equivalent to theorem 3.4 in this case. Proof of theorem 3.4. This follows from remark 3.6.
4. PROOFS OF THE MAIN RESULTS A. ODE Remark 4.1. Let us consider F!$ + Wp’ defined by
the operators
J, F from remark
(S(u), u) = _/fl Iu(P-~u u dt,
2.4 and the operator
S:
for all u, u e fi i.
0
J is a (p - 1)-homeomorphism The operators J, S are (p - l)-homogeneous, ti’p onto W,-? (see definition 2.1) satisfying the strong monotonicity condition (J(u) - J(u), u - u) e CJJU- UJIP,
of
(M)
for all u, u, E F!‘Aand F, S are completely continuous with respect to the completely continuous imbedding F!‘LC Lp. The mappings u * S(u+), u --, S(u-) are completely continuous, too. LEMMA4.1. Let the assumptions of theorem 2.1 be fulfilled. Set H(u, t) = J(u) - tF(u) - tg* - (1 - t) AS(u),
(4.1)
fortE(O,l),uE!Q, where A < A1 and ;1 E (Ai, Ai+J in the case (E,) and (Ei), respectively. Then there exists r > 0 such that H(u, r) f 0,
(4.2)
for all t E (0, l), u E I@; with /lullz r. LEMMA4.2. Let the assumptions of theorem 2.2 be fulfilled. Put H’(u, t) = J(u) - tF(u) - tg* - (1 - t) poS(u+) + (1 - t)~~S(u-),
(4.3)
L. BOCC.ARDOet al
1098
l), u E fij, where U, s ,uOs ,UiL1.V,s ~0 s v,, I are arbitrary but fixed. Then there exists r > 0 such that H’(u, T) # 0, (4.2’)
for T E (0,
for all t E (0, l), 11E IQ, l/u// 2 r. Proof of lemma 4.1. Can be obtained from that of lemma 4.2 by setting &lo= v. = A, writing x, 2 instead of x=, ji and using theorem 3.1 instead of theorem 3.2. Therefore we give only the following proof. Proof of lemma 4.2. Let the assertion be not true. Then there exist u,, r,, (n = I 2,. . .)
such that
and H(u,, t,J = 0, i.e. also
J(u,) -
- (1- t,),u,s(o;) + (1 -
5,
t,)voS(u,)
= 0.
(4.4)
We shall show that it follows from here that v,+
F(lln)
llll,,IIp_l+= h* f(. 94sJ>
in Wi.
v
in Wp!
-;i+luIP-’
(4.5)
for some h” E W,!
v+ - ;i_lo~P-‘v-
in L,,
(4.6)
(4.7)
ll~~nll~-’
(at least for a subsequence) with some j+, ;Z_ satisfying (Ef’). It follows from (4.4) that J(vn) - J(um), v, - u,) =
F(un>
( Ii lIu,llp-l 5
-
-
t
F(G)
ml/%Ap-l ~
(4.8) We can suppose v,,--, v uniformly with respect to the completely continuous imbedding l&k c C((0, A). It follows from here using (G) that the right-hand side in (4.8) tends to zero as n, m+ + x and therefore {u,} is strongly convergent in *i by (M), i.e. (4.5) holds. We have t > 0 because in the opposite case we would obtain by the limiting process from (4.4) that u is a nontrivial solution of (EP,,,,, ) and this would be the contradiction with [y,,, v,,] E A0 (see remark 2.2). Now, (4.6) follows from (4.4), (4.5) because all the other members
1099
Generalization of Fredholm alternative in (4.4)
are convergent.
Further, it follows from (G), (4.5) and the imbedding @j, C C((0, n))
that i”,$!?j is bounded in L, (for q see (G)), i.e. we can suppose
fLUn(J)_h lI4IIIp-l
inL
(4.7’)
4’
for some h E L,. We obtain from (G) that h(t) = 0 a.e. on M0 = {t E (0, n); u(r) = 0) and therefore we can write h(r) = f(t) ]u(~)/P-~ u(t) and put f+(t) = ii(t)
on M+ = {t E (0, n); u(t) 2 O},
f- (1) = a(t)
on M- = {t E (0, n); u(t) 5 0).
The uniform convergence u,--, u and (E,) or (EJ directly imply that ,Y+ and j_ satisfy (Ef=) in the sense of remark 3.1). This together with (4.7’) yields (4.7). Now, using (4.5), (4.6), (4.7) we obtain by the limiting process from (4.4) that u is a nontrivial solution of (EP,+) with x+(t) = Q+(f) + (1 - r),u,, x_(t) = Q_(t) f (1 - t)uO satisfying (El). This is the contradiction with theorem 3.2. Proof of theorem 2.1. Using remark 2.4 it is sufficient to prove the existence of the solution of the equation (AE) which is equivalent (with respect to remark 4.1) to
w + F(J_‘(w))
= g*.
(4.9)
Lemma 4.1 ensures the existence of r such that (4.2) holds, i.e. (4.10)
R(t, w) # 0, for all t E (0, l), w E Wp‘;‘, jIwjI* = R, for R sufficiently large, where R(t, w) = w - rF(J_‘(w))
- (1 - t)/%S(J-i(w))
(We have used the fact that J is (p - 1)-homeomorphism-see It is
- rg*.
remark 4.1).
deg[l-AS(J-‘);B;i,O]fO by Borsuk theorem (see e.g. [12]) b ecause S(J-‘) is odd. (I denotes the identity mapping in W;i). Using (4.10) and the homotopy invariance property of the Leray-Schauder degree, it follows from here deg[Z - F(J-‘)
- g*; BE, 0] # 0.
This implies the existence of the solution of (4.9). Remark
4.2. If [p,,, vo] E A; for some i (for the notation see remark : IV;’ --, IV;’ is defined by deg [Z,,, v,,; Bi , 0] f 0, where ZpO~yO
Z pO,yO(w)= w - ,u@s(J-l(w)+)
+ vJ(J-i(w)-).
2.2. 2.3) then
L. BOCCARDO et al.
1100
This can be proved using Borsuk theorem. the homotopy invariance property of the degree, the fact that the component Ai contains the point [A, A], where A is not an eigenvalue of (EP) and that the operator I - AS(J-‘) is odd. For details see [8]. Proof of theorem 2.2. This is the same as that of theorem 2.1 but we use lemma 4.2 instead of lemma 4.1, the homotopy
H(t, w) = w - tF(J-l(w))
- rg* - (1 - t)pOS(J-l(w)+)
+ (1 - r)VoS(J-‘(wj-)
instead of fi and remark 4.2 instead of Borsuk theorem. LEMMA 4.3. Consider the situation from theorem 2.1 and 2.2 but replace (E,), (EJ and (El) by (EL,), (EL,) and (ELI), respectively. Let A be an odd (p - 1)-homeomorphism of I@; onto Wp;’ which is (p - 1)-quasihomogeneous with respect to J. Define H, H’ as in lemmas 4.1 and 4.2 but with J replaced by A. Then the assertion of lemmas 4.1 and 4.2 in the case of theorems 2.1 and 2.2, respectively, holds. Proof. We shall consider the situation of theorem 2.2 only. The case of theorem 2.1 can be obtained from here by setting ,uo = v. = A and using theorem 3.1 instead of theorem 3.2. Suppose that our assertion is not true. Then there exist IL,,,r,, such that ]lunl\-+ + TJ, r,, E (0, l), r, ---, r, u, = u,,/]Iu,I]- u and
+ (1
- r,),uoS(u,‘)
- (1 - r,)v,S(u,).
(4.11)
It is ]~~(t)l--+ 3~and f07 un 0))
f(b un 0))
= lrln(f),P_ZU,(t) 14)lP-*%0) I141p-2
(n sufficiently large) for a.e. t E M = {t E (0, x); u(t) f 0). It follows from here fk
u&)>
IlUnllP- l
u’(t)- ~_(f)lu(t)p-2U-(t) --, 2+ (~)l~(w-~
(4.12)
a.e. on M, where f(t, s) X? = ,42yx -]sIp-*s’
Simultaneously,
it follows from (G) f(ry u, (0) ---, o
IMP- ’ a.e. on MO = (0, n)\M and therefore (4.12) holds a.e. on (0, ~7).Using Lebesgue’s theorem and (G) we obtain that the convergence in (4.12) takes place also in the Lt-norm. It follows from here (and remark 4.1) that the right-hand side in (4.11) converges in W;* to Q+(u]P-*u+
- rf-lu]P-*o-
+ (1 - t)/.&]P-‘uf
- (1 - T)Va]uIP-*u-
Generalization
of Fredholm
alternative
1101
and therefore
GunI I-+ J(u)
llUnllP -
(see definition 2.1). It is u # 0 because
according to the assumption that A is (p - l)-homeomorphism. solution of (EP,=) with x+(t) = rX+(t) + (1 - r)&,X-(t) satisfying (E;‘).
This contradicts
= Q-(t)
Hence, we have a nontrivial + (1 - r>vlJ
to theorem 3.2.
Proofofrheorem 2.3. This is the same as that of theorems 2.1, 2.2 but we use lemma 4.3 instead of lemmas 4.1, 4.2.
LEMMA 4.4. The assertion of lemma 4.3 remains true also under the assumptions (E,), (E,) and (El) if (G) is satisfied with
(;++
4=p* Proof. Arguing via contradiction
1).
we obtain again (4.11). The condition (G) implies, now,
that F(u,)
II4~-* is bounded in Lp.. We obtain from here (using completely continuous imbedding L,. c IV,-?)
F(un>
IIu,Ip-l--,h* in IV,-? for some h” E IV>‘. Analogously as in the proof of lemma 4.2 we obtain (4.7) (with q = p*) which implies that the right-hand side in (4.11) converges in WP;’ again to r~+JuIp-*u+ - r~-/u]p-*u-
+ (1 - t) /&)Iu]P-%+ - (1 - t)Y&IP-*U-.
To finish the proof we argue as in the proof of lemma 4.3. Proof of theorem 2.4. This is the same as that of theorems 2.1, 2.2 using lemma 4.4 instead
of lemmas 4.1, 4.2. Remark 4.3. For the proof of theorem 2.1 under a more restrictive assumption (G) with q = p*
and a special version of theorem 2.4 see [9]. B. PDE
LEMMA 4.5. Let the assumptions of theorem 2.5 be fulfilled and define H as in lemma 4.1 but with J, F, S from remark 2.7. Then the assertion of lemma 4.1 holds.
1102
L. BOCURDO
er al
Proof. The case (E,) is the same as in lemma 4.1 but we use theorem 3.3 instead of theorem 3.1. Consider the case (E,) and suppose that our assertion is not true. Then there exist T,, u,, such that
and
-
J(un>- tn Analogously here
as in the proof
of lemma
(1 - t,)AS(o,)
4.1 (but using (G’) instead
= 0. of (G))
(4.13) it follows
from
, u,+
F(un)
,,u.,,P_, + h*
u
in
.
in IV;’
f(.AlW IMlp-’
kVL
(4.14)
for some h” E W;‘,
-
~IuIP-~ u in L,.
(4.16)
with Ix(t) - AI 5 C~ - 6 a.e. on R. This implies
F(un> -II4w’ Hence,
AS(u,) +
(x(t) - A) lu(t)iP-%(t)w(t) dt d c; - 6.
we obtain
for n sufficiently
large and this is the contradiction.
Proof of theorem 2.5. This follows from lemma 4.5 by the same way as that of theorem lemma 4.1.
2.1 from
Proof of theorem 2.6. Combining the considerations from the proofs of lemmas 4.4. 1.5 it is easy to see that an analog of these lemmas for the situation from theorem 2.6 holds. The remainder of the proof is clear from the previous. Proof of theorem 2.7. This is the same as that of theorem on theorem 3.4 must be used. Acknowkdgemenrs-The
authors
would likr to thank
Luca Leonori
2.1 but an analog of lemma -1.1 based
for the figures.
REFERENCES
1. AMBROSEITI invertible
A.
&
M.-\NCINI
linear part. Annali
G.,
Theorems
Scrc. norm.
of existence and multiplicity sup. Piss 5. 15-28 (1975).
for nonlinear
elliptic
problems
lvith non
Generalization of Fredholm alternative
1103
2. AXXNN H. & ZEHNDER
E., Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations Annali Pisa VII, 539-603 (1980). 3. CASTROA., A semilinear Dirichlet problem, Can. /. Math. 31,337-340 (1979). 4. CASTROA. & LAZERA. C., Critical point theory and the number of solutions of a nonlinear Dirichlet problem Annali Mat. pura appl. 120, 113-137 (1979). 5. CODDINGTOX E. A. & LEVISSONN., Theory of Ordinary Differenrial Equarions, McGraw-Hill, New York (1955). 6. DE FIGUEIREDOD. G., The Dirichlet problem for nonlinear elliptic equations: a Hilbert space approach, Lecture ‘Votes in Mafhematics 446. 14&16j, Springer, Berlin (1975). 7. DOLPHC. L.. Nonlinear integral equations of the Hammerstein type. Trans. Am. math. Sot. 66.289-307 (1949). 8. DRABEKP., Ranges of a-homogeneous operators and their perturbations. Gas. p&f. mar. 105, 167-183 (1980). 9. DR~BEK P., Solvability of boundary value problems with homogeneous ordinary differential operator. Rc. Uniu. Triesfe (to appear). 10. FL’i%, Soloability of Nonlinear Equarions and Boundary Value Problems. D. Riedel. Holland (1980). 11. FCUK S. & KUMER A., Nonlinear Differential Equadons, Elsevier. Amsterdam (1980). 12. FuCiK S., NE~AS J., SOU~EK J. & SOU~EKV., Spectral analysis of nonlinear operators, Lecrure iVoles in Malhemarics 346, Springer. Berlin (1973). 13. KANNANR. & LOCKERJ., On a class of nonlinear boundary value problems, J. diff Eqns 26. l-8 (1977). 14. KAZDA.VJ. & WARNERF.. Remarks on some quasi-linear elliptic equations. Communs pure uppl. ,Cfarh. 28, 567597 (1975). 15. LANDESMASE. M. & LAZERA. C., Linear eigenvalues and a nonlinear boundary value problem. Pacif, J. Marh. 33, 311-328 (1970). 16. LAZERA. C. & LEACHD. E., On a nonlinear two-point boundary value problem, 1. marh. Analysis Applic. 26.2027 (1969). 17. hlhWHlN J.. Topological degree methods in nonlinear boundary value problems, CBMS Regional Conference Series in Mathematics, No. 40, Am. Math. Sot., Providence, RI (1979). 18. MAWHIN1. & WARDJ. R.. Nonresonance and existence for nonlinear elliptic boundary value problems. Nonlinear Analysis 6,677-684 (1981). 19. N&AS J., Sur I’alternative de Fredholm pour les operateurs non-lineaires avec applications aux problemes aux limites. Annali Scu. norm. sap. Pisu 23, 331-345 (1969). 20. POCHO~AJEV S. I., On the solvability of nonlinear equations involving odd operators. Funkcional.Anal. i Priloieniju 1. 66-72 (1967). (In Russian.) 21. THEWS K., A reduction method for some nonlinear Dirichlet problems. .VonlinearAnulysis 3. 785-813 (1979). 22. WILLIAMS S.. A nonlinear elliptic boundary value problem, Pacif. J. .!lurh. 4, 767-774 (1973).
Vol
10. No.
10. pp. 1083-1103.
19%.
0362-5%X,‘86 P3.M) - .M) Perqamon Journals Ltd.
GENERALIZATION OF FREDHOLM ALTERNATIVE NONLINEAR DIFFERENTIAL OPERATORS
Department
FOR
Lucre BOCCARDO Universita di Roma “La Sapienza”, P. A. Moro 2, 00185 Roma. Italy
of Mathematics,
PAVEL DRABEK Department
of Mathematics, Technical University of Plzeti, Nejedleho sady 14, 30614 Plzeti, Czechoslovakia DANIELA
Department
of Mathematics,
GIACHETTI
II Universita di Roma, Via Orazio Raimondo. 00173 Roma. Italy
and MILAN
KUCERA
Mathematical Institute of Czechoslovak Academy of Sciences, iitna 25, 11567 Praha 1, Czechoslovakia (Received for publication 21 October 1985) Key words and phrases: Nonlinear noncoercive equations, nonlinear eigenvalues, Fredholm alternative
1. INTRODUCTION WE SHALL
consider the boundary value problem
-(Iu’(t)lP-* u’(t))’ =f(t, u(t)) + g(r), t E (0, ‘2). u(0) = u(n) = 0.
I
It will be proved that if
lie (roughly speaking) between eigenvalues of the eigenvalue problem - (Iu’IP-~u’)’ = AIulP-*u
on
(0,x) (EP)
u(0) = U(ir) = 0
1
then (BP) is solvable for anyg E L, (theorem 2.1). Moreover, certain more general assumptions about f under which (BP) is solvable will be formulated using properties of the problem - (Iu’IP-‘u’)’ = ,uIuIP-~u+ - ~luIp-~u_
on(0, x) (EP,, y)
u(0) = u(n) = 0
1
(theorem 2.2). Further, we shall prove under the similar assumptions about f the solvability of the equations asymptotically close to (BP) (theorem 2.3). Certain weaker results will be obtained for partial differential equations of the analogous type (theorems 2.4-2.6). Let us remark that for the special case of differential operators under 1083
1084
L. BOCCARDO et al.
the consideration our results represent generalizations of abstract Fredholm alternative for nonlinear operators proved by NeEas [19], Pochoiajev [20] and developed by many authors (see [12] for further references). Much work has been devoted to the study of differential or abstract nonresonant semilinear equations of the form Lu + Nu = h with L linear and N nonlinear operator In the differential case
(see e.g. [lo, 2, 171).
LJA= - ,i;.F<, (-l)“‘D’($j(x)D’u)
=f(x, n),
existence and uniqueness results are given by Dolph [7], in the framework of integral equations and by Landesman and Lazer [lj], Lazer and Leach (161, Kannan and Locker [13]. Other results are given by Williams [22], de Figueiredo [6], Kazdan and Warner [14], Thews [21], Castro [3], Castro and Lazer [4], Ambrosetti and Mancini [l], provided there exist real numbers p, q such that A
11) q < l.,t,.
--s
I
U
The general case fb. pi ~ p(.r) ~ 11
u>s q(x)
< A,_,
has been studied by Mawhin and Ward [18]. However Lazer and Leach showed in [16] that the previous inequalities cannot be replaced by
A.
11
‘171.
In our problem the principal part also is nonlinear. generalization of the previous ones.
Therefore,
our results are a partial
Notarion. Let p 2 2. We shall denote by I@; the usual Sobolev space on (0, X) or on Q (the domain in Rv), Wp;’ its dual space, (. , .) the pairing between I8’; and Wp’_?.The norm in I@. and in W;l will be denoted by I].// and ]I.jj*, respectively. The symbol B: will denote the ball in Wp’ with the radius r centred at the origin, deg [T: B: , 0] will be the Leray-Schauder degree of the mapping T: Wp’_?---f IV;’ at the origin with respect to B:. The strong convergence and the weak convergence will be denoted by + and -, respectively. We shall denote by & (i = 1,2, . . .) the eigenvalues of (EP). It is known that the eigenvalues of (EP) form an increasing sequence {A,}of positive numbers, An-, + r: (see [12, 81 and remark 3.2). By the solution of a boundary value problem for ODE we shall always mean a function u E C’((0, x)) satisfying the corresponding boundary conditions such that Iu’~J’-‘u’ is absolutely continuous on (0, n) and the corresponding equation holds a.e. in (0, n). By the solution of an initial value probfem on (to, T) we mean a function u E C’((to, T)) satisfying the initial conditions such that ]u’IP-%’ is absolutely continuous on compact subintervals of (r~, T) and satisfying the corresponding equation a.e. on (to, T).
Generalization
1085
of Fredholm alternative
The positive and negative part of u will be denoted by U- and u-, respectively. The function f is supposed to satisfy Caratheodory’s conditions automatically. 2. FORMULATION
OF IMAIN RESULTS
AND SOME
COMMENTS
A. Ordinary
(Eo) ’ S_+r IsIp-Zs c “~S~P
-IsIp-zs
s *i+t
- 6
for a.e. t E (0, n) with some 6 > 0 and i 2 1, where A, (n = 1,2. . . .) are the eigenvalues of (EP). Then (BP) has for each g E Lr at least one solution. Remark 2.1. Let K be the square from Fig. 1 and Fig. 2, respectively. respectively, can be written as
litn~;p#$)-,I__~p~] S
E K,
S
Then (E,) and (E,),
fora.e.tE(O,n) e>
lim inff(t' s_+* IsIP-ts'
liminf I__I
f(tJ' IsIp-2s
E I
K
'
fora.e.tE
(O,j-o.
Further we shall describe more general set K such that (E) guarantees (BP).
Fig. 1
the solvability of
L. BOCCARDOer al.
1086
Fig. 2.
Remark 2.2. Denote
by A-, the set of all couples [p, V] E IL!*such that the problem (EP,,.) has a nontrivial solution. It is proved in [8] that [II, V] E A_, if and only if one of the following conditions is satisfied (see also remark 3.5): (i) p = Al, v arbitrary (A,, are eigenvalues of (EP)); (ii) p arbitrary, v = Al; (iii) p > Al, v > Al and at least one of the numbers w,(p, v), w,(p, v), W&L, u) is positive integer, where VP Wl(lA
v)
=
Wj(,U,
v)
=
(pl,pk;
. v’lP V”P)(A,)“P
(vVP
-
~yP)y’iP
VP
+
v UP
( ,U
Fig. 3.
)A
’
1’”
’
Generalization of Fredholm alternative
1087
The set consists of the lines intersecting the axis ,U= Y at [I.*,,I,] (i = 1,2,. . .), see Fig. 3. Put A, = IW\A_,. It is proved in [8] that if [,u, V] belongs to any component of A0 containing some [j_, A] (in this case it is A # Ai, i = 1,2,3, . . .; see Fig. 3) then for any g E L1 there exists a solution of the boundary value problem = ~~KIP-*K+ - ujc~lP-*~- + g
-(IK’IP-*K’)’ U(0) =
427)
=
on (0, x), (BPU.“)
0
(cf. remark 4.2). (Particularly, for any ,UE R, even if p = A,, there are v such that (BP,..) is solvable with any g E L’). This can be understood as a generalization of the Fredholm alternative for the problem (BP) withf(t, U) = $@-*u (for Fredholm alternative for nonlinear equations see e.g. [12])*. The following theorem 2.2 is a generalization of this existence result as well as of theorem 2.1. THEOREM 2.2. Suppose that f satisfies (G) and that there exists a component AL of A0 containing a point of the type [A, A] and points [,u,, vi], [lu,il, vicl] E Ai such that
for a.e. t E (0, n). Then (BP) has for any g E L, at least one solution. Remark 2.3. The index i has a precise sense in theorem 2.2. if we denote by Ai andAd (i = 1,2,. . .) the component of A,, containing a point [I, A] with A < At and ,I E (&, I.;+,),
respectively (see Fig. 3). In the case i = 0 the estimate from below in (EA) is not necessary (i.e. we can formally put y, = v. = - x); this case is the same as (E,) in theorem 2.1. The condition (E/) for i 2 1 is a generalization of (Ei) which corresponds to the case !ci = I/i = pi + 6, pi+1 = Vi+, = I-i+1 - 6. The condition (El) is equivalent to (I?) from remark 2.1 with the oblong K = {[v, nz] E W; /J E (,Ui,,~i+l), v E (Vi, Vi+r)} (see Fig. 3, cf. remark 2.1). Remark 2.4. Let J, F: I@; + W,! (J(u),
be operators
such that
u) = _/x ju’JP-2u’u’dt,
for all
K,
u E
I&’
b,
0
(F(u),
u) = \“f(r.
u(t))u(t)
dt,
for all
K, u E @ L.
0
The assumption (G) guarantees that F is well defined. (The proofs in Section 4 nork also with q = 1 in (G) if we use the fact that bounded equi-integrable subsets of L1 are weakly * Let us realize that (BP,..) can be written as (BP) withf(t, u) = ,~lulp-~u, u > O,f(t. u) = vIuIp-‘u, u C 0.
L. BOCCARDO~~~~.
compact instead associate g’ E W,-? defined by
L,. q > 1.) To an arbitrary g E L1 we shall
the reflexivity
(g*, u) = 1” g(t)u(r) dt,
l&k+;.
for all
0
If u E k!‘i is a solution of the abstract equation J(u) - F(U) = g*,
(AE)
then u is the solution of (BP) (in the sense mentioned in Notation, Section 1). This can be shown using the standard regularity argument for ordinary differential equations (see e.g. [8]). Remark 2.5. If f(t, U) = A]u]P-*u, where A is not an eigenvalue
of (EP) then the assertion of theorem 2.1 follows from abstract Fredholm alternative for nonlinear operators (see e.g. [12, 19, 201). Following the ideas from [12, 191 we shall generalize theorems 2.1, 2.2 for the equations asymptotically close to (AE).
Definition 2.1.
Let A0 : I@; + W>’ b e an a-homogeneous operator (a > l), i.e. Ao(tu) = uE I@;. Then A: I!‘;W;! is said to be a-quasihomogeneous with
t”AO(u), for all t>O, respect to A0 if
t,-+O,u,-u,t;A
-+g*
in
WP;‘,
imply g” = A,(u). The operator A is said to be an a-homeomorphism
of I!‘; onto Wpl if it is a homeomorphism
of fi; onto Wp;’ and there exist Ci, C2 > 0 such that C,IIuII” s ItAut]*s C21141'
(2.1)
for all u E tii. Further, we shall suppose that there exist
. f(t9s) ,!F%IsIp-2s and consider the estimates
,l&lx
f(t, s>
5s Al
IsIp-2s
f(tt s>
- 6, ,“y
szAl - 6;
lslp_*s
f(4 s)
f(t73)
Ai + 6 6 s_+x lim -~~~~~~~ s ki+i - 67 Ai f(t,
pi
a.e. in (0, Id).
G
s)
W-o)
+
6
s
,gyz
f(4
IsIp-*s
s>
s
Ai+l
vi G I__x lim -IsIP-*s -= Vi+‘,
-
6;
W-J
(EL,‘)
1089
Generalization of Fredholm alternative
2.3. Consider the situation from theorem 2.1 and 2.2 but replace (E,), (E,) and (El) by (H-c,), (EL,) and (EL!), respectively. Let A be an odd (p - 1)-homeomorphism of I@; onto W>’ which is (p - l)-quasihomogeneous with respect to J. Then for any g* E Wp;’ there exists at least one solution of THEOREM
A(u) - F(u) = g*.
(A%)
that q =p* (for q > 1 see condition (El). Then the assertion of theorem 2.3 remains true.
THEOREM 2.4. Let us suppose
(G)),
(E,),
(Ei) and
Remark 2.6. The proofs of theorems
2.1-2.4 (see Section 4) are based on the properties of the Leray-Schauder degree and on some results concerning the corresponding homogeneous equations formulated and proved in Section 3. The proofs of these results in the cases (E,), (E; ) or (EL,), (EL; ) for i 2 1 are based on the method which cannot be used for partial differential equations. Therefore the results formulated further for PDE are weaker than previous ones for ODE. B. Partial differential equations
In this part we shall consider a bounded domain R C RN with a Lipschitzian boundary JR, the boundary value problem
u=OonaS2,
and the eigenvalue problem
=0 Remark 2.7. Let us define operators
J, S:
on lf’;-W,;
1 an. by
(J(u), v) =
forallu,vEW~andletF:W~+W;!,g*EW; ? be defined in the same way as in remark 2.4 but by means of integrals over 52. The growth condition (G) will be replaced by jf(X, $1 Z m(x) + cjs/p-‘, for all s E R, a.e. x E Q, with some mE
L,.
(i+$=
l),c>O,
(G’)
L.Bocca~oer al.
1090
which guarantees that F is well defined and continuous. (A more general assumption would be sufficient for the continuity of F, see [ll]; but we shall need (G’) in the proofs in Section 4.) The operators F, S are completely continuous with respect to the completely continuous imbedding fik C L,. A function u E fij is said to be a weak solution of (B?) if it satisfies J(u) - F(u) = g*. Analogously,
the weak solution of (fi)
(AE)
for a given J. is the function u satisfying
J(U) = AS(u). A number J. is said to be an eigenuafue of (ET) if there exists the corresponding nontrivial weak solution (the eigenfunction) of (l?P). It is well known that the Ljusternik-ScJhnirelmann theory ensures the existence of an infinite sequence of positive eigenvalues of (EP), but this theory does not give all eigenvalues in general (see e.g. [12]). The set of all eigenvalues can be more complicated and we cannot speak about the ith eigenvalue in the following theorems because we must consider all eigenvalues, not only those obtained by Ljusternik-Schnirelmann theory. However, there exists the least eigenvalue Ai > 0. THEOREM
2.5. Suppose that f satisfies (G’) and one of the following estimates: li,m_~~p$~*, i. - ci + 6 5
-6
S
f(X>s)
lim J-+Ix lsp-2s’
lim
f0ra.e.
xER;
0%)
sup f(X> s) S_.zr jsp-2s S A + ck - 6
(E).)
for a.e. x E Q; with some A which is not an eigenvalue Then for any
of (l?P), 6 > 0, where C~ = ,_$i, Mu) -
m4ll*
there exists at least one weak solution of (KP). Remark 2.8. It is easy to see that CA> 0 if 13.is not an eigenvalue
continuous
because S is completely
(remark 2.7).
2.6. Let A be an odd (p - l)-homeomorphism of t&‘i onto W;’ which is (p - l)quasihomogeneous with respect to J. Suppose that f satisfies (G’), (E,) and (E,). Then for each g’ E WP;’ there exists at least one solution of
THEOREM
A(u) - F(u) = g*. Remark 2.9. The case (E,) in theorems
(A%)
2.5, 2.6 is precisely the same as in theorem 2.1-2.4 for ODE because the corresponding information about the homogeneous equation (theorem 3.3) can be obtained using the variational characterization of the first eigenvalue. This is not
Generalization
of Fredholm
1091
alternative
possible in the case (Ei), (EL,), for i 2 1, p > 2 (cf. remark 2.6) and therefore the validity of an analog of theorems 2.1-2.4 for the case of PDE seems to be an open problem. In the case p = 2 variational approach can be used even in the case i 2 1 (see remark 3.6) and therefore the following theorem holds. 2.7. If p = 2 then the assertion of theorem 2.1 and the assertions of theoremsz.3, 2.4 in the case of the situation of theorem 2.1 holds with (BP), (EP) replaced by (KP), (EP).
THEOREM
Remark 2.10. We cannot speak about an analog of theorem 2.2 or of the second case in theorems 2.3, 2.4 (i.e. assuming (E/ ), (ELI )) b ecause vve have no corresponding information about the problem (EP,_) for the partial differential case. Remark 2.11. We are going now to explain a different procedure in order to prove the existence of solutions of the problem (BP), under the hypotheses of theorem 2.2. The approach is quite different from the previous one since no argument from degree theory is used and the solution is found by approximation. ru’evertheless, both the proofs use the “shooting method” (see theorem 3.1) as an essential tool. Let us only sketch the main steps of the proof. (i) We define U, as a solution of the equation
-(IU’(t)p-V(t))’
=fn(f, u(t)) + g(t).
t E (0, ;r)
u(0) = u(x) = 0, where fn(t, s) = S~uil~IJ’-‘~ + ~$,(t, s),
and T,, is the truncation operator at n. The existence of such a solution is guaranteed by a theorem of [12]. (ii) The sequence (~,),,~~v defined in (i) is bounded in fij. The proof is obtained by contradiction, using the “shooting method”. (iii) The sequence (~,),,~.v is compact in I&k. The cluster points are solutions of the equation (BP). 3. INVESTIGATION
OF THE
CORRESPONDING
HOMOGENEOUS
EQU,\TIOXS
In this section we shall prove some results which will be the bases for the proof of theorems 2.1-2.7. A. ODE of the second order with constant coeficients First of all, we shall consider the special case (theorem 3.1) which is sufficient for the proof of theorem 2.1 and only then we shall formulate more general theorem 3.2 which will be used for the proof of theorems 2.2, 2.3, 2.4.
L. BOCCARDO er al.
1092 THEOREM
3.1. Let x E L i satisfy one of the following estimates: x(t) 5 A1 - 6
a.e. on
Ai + 6~X(t)~~i+l
- 6
(0, j-r),
a.e.on
(Et)
(0,X),
(E?)
with some 6 > 0, where A, are the eigenvalues of (EP). Then there is no nontrivial solution of -+4-2w’)’
= xIwIP-Q$J
on
(0,~)
W(0) = W(Jr) = 0.
(EP,) 1
THEOREM 3.2. Let x+, x- E L, and let there exist a component Ai (for the notation see remarks 2.2, 2.3) of A0 containing a point of the type [A, A] and points [,Ui,vi], vi+ ,] E AL such that [Pi+l9 pi S x+(t) S pitI,
Vi 5 x_(t) s vi+1
a.e. on
(E;=)
(0, n).
Then there is no nontrivial solution of the problem -(Iw’IP-~w’)’
= ~+IwIp-~w*
- ~_IwIp-~w-
on
(0,x),
W’, )
w(0) = w(n) = 0.
1
Remark 3.1. If A& are defined in remark 2.3 then in the case (Ei’)
the estimate from below is not necessary, i.e. we can set formally p0 = v0 = - = (cf. remark 2.2). B. PDE THEOREM 3.3. Let x E Z+JS2) satisfy (Ei), there is no nontrivial weak solution of
where A, is the least eigenvalue
of (E?P). Then
THEOREM 3.4. (Linear case.) If p = 2 then the assertion of theorem 3.1 holds with (EP,) and (EP) replaced by (EP,) and (EP), respectively. Remark 3.2. The proof of theorem 3.1 in the case (Ei) is based on the variational characterization of A, and this method can be used also for PDE (theorem 3.3). An analogous approach for i 2 1 can be used only in the linear case p = 2 and only for this case we are able to prove the complete analog of theorem 3.1 for PDE (theorem 3.4), cf. remark 3.6. But in the case (Ef), i I 1, p > 2 the proof of theorem 3.1 will be based on the comparison lemma 3.1 and this will be proved by shooting method which cannot be used for PDE.
1093
Generalization of Fredholm alternative
Remark 3.3. For the proof of theorems 3.1 and 3.2 we shall use the properties
of the following
two initial value problems -(Iq-*~‘)’
= +JP-*u,
ll(t0) = 0, u’(to) =
(Y,
I
UPi)
and -(Iw’I@w’)’
=
xlwlp-‘w,)
WJ
1
w(hl>= 0, w’(h)) = p,
respectively, where to, A E R and x E L,(to, T) for some T > to, x(t) Z r z=-0 a.e. on (to, 7). The existence of the solution can be proved by standard methods (see e.g. [j]) and it is not hard to show that the solution is uniquely determined [8]. Further, it follows directly from the equation (IP,) that the solution is concave and convex on the intervals where w > 0 and w < 0, respectively. Particularly, this is true for (IP,) with A > 0. LEMMA 3.1. Let ui and u2 be the solutions* of (IP,,)
and (IP,,),
respectively.
with some to,
A,, AZ, (YE R, A,, AZ> 0. Set t& = inf{t > r,; u;(t) = 0). Suppose that x E Li(to, r), t,~ (to, T), where
w is the solution cf (IP,) on (fo, T) with /3 E W, (u/3> 0, and t, = inf{t E (to, T); w(t) = 0).
If Ai ~5x(t) s A2 a.e. on (to, T) then fl, Z t, ZGtA,. Proof. We can suppose (Y> 0, /3 > 0 without loss of generality.
Let us prove fz E ti,. We shall write A and u instead of A2 and 11~for the brevity. There are uniquely determined points ri E (to, tl) t; E (to, fx) such that u’(ti) = 0, w’(t;) = 0, u’(t) < 0 w’(f) < 0
on
on
(tf, tA), (3.1)
(tk, fX)
(see remark 3.3). The function G(t) = u(t - t; + ti) is the solution of -(]$(p-2$)’
= j+(P-Q, (R)
Lz(t, + t; - t;) = 0, fi’(r, + t; - ti) = CC. We shall prove that ti.+t’-tl,st x
X’
(3.2)
* In the sense mentioned in Notation (Section 1); remark that U,are defined on (r,,, + =) and that rL,are finite numbers, precisely see remark 3.4.
L. BOCCARDO er al.
109-I
Suppose the contrary (see Fig. 4). It is (w/il)(t;) > 0, (w/ti)(t,) = 0 and therefore there exists TV (t;, tx) such that (w/c?)‘(t) < 0, i.e. (w’U - WC’)(~) < 0, i.e. also F(t) CO”, where F(t) = (~w’]p-%‘(ti)p-’
- (W)P-i]ti’IP-%‘)(t).
Simultaneously, F(t;) = 0 because w’(t;) = 0 = ti’(r;) and therefore A C (t;, t) of positive measure such that F(t) < 0, F’(t) < 0
there exists a set
on A.
(3.3)
We can write F’(t) = F,(t) + F2(f). where F,(t) = (iw’lp-‘w’)‘(ti)p-’
- (W)“-‘(Iil’I~-~ti’)‘](t).
F?(t) = (p - l)(w’~‘(Iw’jp-~up-?
-
iti’lp-2~~~-2))(f).
It follows from (3.3)* that (w’ic - WC’)(t) < 0 on A and this together with (3.1) gives fEA.
F?(f) > 0, Further.
(3.4)
(IP,), (IP,) imply F,(t) = (A - ~(t))W(t)%i(t)~-l
2 0
a.e. on (0, X)
(3.5)
and (3.4). (3.5) contradicts to (3.3). Hence (3.2) is proved. Now by analogous considerations as above but on the interval (to + t; - ti, r;) instead of (t;, tA + t; - ti) we can prove t, < to + t; - r; and this together with (3.2) implies f, 2 t;, = fA,. The inequality fX 5 tA, can be proved by the analogous considerations. Remark 3.4. If uI is the solution of (IPJ with (Y> 0, A > 0 then u:, is 2(A,/A)“P ,r-periodic function having isolated zero points
rf=fo+j
A, T
i
VP
)
n,(j=0,1,2
)... ),
and lu~(r~)i = N. It follows from elementary investigation of (IP,) (cf. [S]). We can norm all eigenfunctions of (EP) in such a way that u’(O) = 1 because u’(0) # 0 for any nontrivial solution of (IP,) with respect to the unicity of the solutions (remark 3.3). Then A is an eigenvalue of (EP) with the corresponding eigenfunction uE.if any only if uj. is the solution of (IP,) with to = 0, LY= 1 satisfying u*(X) = 0. * We
use
the fact that C(r), w(r) > 0 on (t;, rx) and the function z--* IzIp-*z is increasing.
1095
Generalization of Fredholm alternative
follows from here that all eigenvalues of (EP) are of the form Ai = iP A, (i = 1, 2, . . .) that tfl = I-C.If A c Ai and A > Ai then tt > n and t” c x, respectively, where tf denotes the ith zero point of the solution of (IP,) with t, = 0, LY> 0. It and
Proof of theorem 3.1. Consider the case (Ei). If w is a nontrivial solution of (EP,) then we obtain (multiplying (EP,) by w and integrating by parts)
IK
Iw’IP dt
1=5
I0
.
(3.6)
~1wIP dt
It is well known that
(3.7)
and this is the contradiction because (3.6), (3.7) cannot hold simultaneously under the assumption (E;). Consider the case (El), i 2 1. Let Di and Vi+1 be the solution of (IPA) with i = i.i + 6 and A = /I,+1 - 6, respectively, and with (Y= 1. Denote by . ‘;=I
A,
VP
( 1 A,+6
n
t’+l 91
-j
(Ai+l1-
J”
n
o’= 1,2,. . .) the zero points of Uiand u.r+l, respectively (see remark 3.4). If w is a nontrivial solution of (EP,) then we can suppose w’(0) > 0 with respect to the unicity of the solution of (IP,). Hence w is the solution of (IP,) with to = 0, p > 0 and it is easy to see (using lemma 3.1) that t~+‘Stj+Stj
,...,
i),
where t,? denotes the jth (from the left) zero point of w (see remark 3.4). (It follows directly for j = 1; further, we can use lemma 3.1 at to = t; successively for j = 1, 2, . . . , i - 1 for the comparison of zero points of ui(t) = Ui(t - tr + $), W, Ui+,(t) = Ui+l(t - t,?’ + $‘I).) NOW, it follows from lemma 3.1 that w has no further zero point in (tr, n) because tiz] > ,7 (remark 3.4) and therefore w(x) # 0 which contradicts to the boundary conditions in (EP,). Remark 3.5. For the proof of theorem 3.2 we shall consider the initial value problem -(]4P--zu’)’
= pluIP-*u+
-
vIuIP-~u-, 1,
u(0) = O,u'(O) = y
Pu.“1
J
with y, Y, y E R, y # 0. Using remark 3.4 we can realize that u is the solution of (IP,.,) with y # 0 if and only if u is the solution of (IP,) with A = ,u, (Y= I;/( and with A = V, rr = -jyI on
1096
L. BOCCARDOer al.
any interval (to, T) where u is positive and negative, respectively. unicity of the solution of (IP,.,) and therefore nontrivial solutions Any nontrivial solution can be multiplied by positive constant such -1 and therefore we can consider the cases y = 1, y = -1 only. It that the solution u of (IP,,,) is periodic with the period (C)l”
+ (2)‘lP)
Particularly we have the can exist for y f 0 only. that u’(0) = 1 or u’(O) = follows from remark 3.4
n.
The solution UT” of (IP,U,,) with y = 1 has zero points f$” given by
The zero points rf:” of the solution uF” of (IP,U,,) with y = - 1 are given by analogous formulas but the role of ,u, Y in the right-hand side is interchanged. If ~1, v are given then a nontrivial solution of (EP,. “) exists if and only if UC” (‘2) = 0 or u?“(n) = 0 and this is fulfilled precisely for [,u, V] E A-, (see remark 2.2). More precisely, it is t,y;” = nor r;:” = n for [p, V] E A’_,, where Ai, is the component of A_1 containing [Ai, Ai] (Fig. 3). If [,u, V] E Ai (see remarks 2.2, 2.3) then t,r;” < X, ly:” < n, ttJi,+ > .z, f$Y,,- > n. Proof of theorem 3.2. Suppose that w is a nontrivial solution of (EPA=). Let u, and ui+, be the solutions of (IP,.,) with !L = yi, u = vi and ,~l= pi+i, v = vi+,, respectively, and with y = sign w’(O) (it is w’(O) # 0 with respect to the uniqueness). Denote by rj and ff*’ (j = 0, 1,2,. . .) the jth (from the left) zero point of ui and ui+i, respectively. The function w is the solution of (IP,) with x = x1 and x = x- on any interval where w > 0 and w < 0, respectively. It is easy to see from lemma 3.1 and remark 3.5 that
t~+‘5f,~5rj
,...,
i),
(3.8)
where tr is the jth (from the left) zero point of w. (It follows directly from lemma 3.1 for j = 1; further, we can use lemma 3.1 at to = ty successively with j = 1,2,. . . , i - 1 for the comparison of the 0’ + 1)st zero points of u,(t) = Ui(t - ty + ti), W, uitl = u,_,(f - t,? + ti”).) Now, using lemma 3.1 at tr once more, we obtain that w has no zero point in (tr, n) because t;:f > n (see remark 3.5). Hence, w(n) # 0 which is the contradiction. Proof of theorem 3.3 is the same as the first part of the proof of theorem 3.1 (the case (E,)). Remark 3.6. On the basis of the variational characterization of LS-eigenvalues (i.e. eigenvalues which can be obtained by Ljusternik-Schnirelmann theory, see e.g. [12]) it is possible to prove the following.
1097
Generalizationof Fredholmalternative COMPARISONLEMMA.If AAand A: is the nth LS-eigenvalue
of the problem
-~&(~~~p-2~) =AxIuI~-~u 1
I
I
u=O
onS2, 0naR
(*) I
with x = xl and x = x2, respectively, and x1, x2 E Lx(Q), x, s x2 a.e. in Q, then On the basis of this comparison lemma the following assertion can be proved. PROPOSITION.If x E L,, pi S x (x) C picl a.e. on R, where yi, pi+l are such that there is no eigenvalue of (I?P) in (pi, pi+l), then A = 1 is not a LS-eigenvalue of (*). But we cannot use this assertion instead of theorem 3.1 in the case p > 2 because it does not e?clude the possibility that A = 1 is (not LS) eigenvalue of (*), i.e. the nontrivial solution of (EP) can exist. In the case p = 2 there are only LS eigenvalues and therefore the mentioned proposition is equivalent to theorem 3.4 in this case. Proof of theorem 3.4. This follows from remark 3.6.
4. PROOFS OF THE MAIN RESULTS A. ODE Remark 4.1. Let us consider F!$ + Wp’ defined by
the operators
J, F from remark
(S(u), u) = _/fl Iu(P-~u u dt,
2.4 and the operator
S:
for all u, u e fi i.
0
J is a (p - 1)-homeomorphism The operators J, S are (p - l)-homogeneous, ti’p onto W,-? (see definition 2.1) satisfying the strong monotonicity condition (J(u) - J(u), u - u) e CJJU- UJIP,
of
(M)
for all u, u, E F!‘Aand F, S are completely continuous with respect to the completely continuous imbedding F!‘LC Lp. The mappings u * S(u+), u --, S(u-) are completely continuous, too. LEMMA4.1. Let the assumptions of theorem 2.1 be fulfilled. Set H(u, t) = J(u) - tF(u) - tg* - (1 - t) AS(u),
(4.1)
fortE(O,l),uE!Q, where A < A1 and ;1 E (Ai, Ai+J in the case (E,) and (Ei), respectively. Then there exists r > 0 such that H(u, r) f 0,
(4.2)
for all t E (0, l), u E I@; with /lullz r. LEMMA4.2. Let the assumptions of theorem 2.2 be fulfilled. Put H’(u, t) = J(u) - tF(u) - tg* - (1 - t) poS(u+) + (1 - t)~~S(u-),
(4.3)
L. BOCC.ARDOet al
1098
l), u E fij, where U, s ,uOs ,UiL1.V,s ~0 s v,, I are arbitrary but fixed. Then there exists r > 0 such that H’(u, T) # 0, (4.2’)
for T E (0,
for all t E (0, l), 11E IQ, l/u// 2 r. Proof of lemma 4.1. Can be obtained from that of lemma 4.2 by setting &lo= v. = A, writing x, 2 instead of x=, ji and using theorem 3.1 instead of theorem 3.2. Therefore we give only the following proof. Proof of lemma 4.2. Let the assertion be not true. Then there exist u,, r,, (n = I 2,. . .)
such that
and H(u,, t,J = 0, i.e. also
J(u,) -
- (1- t,),u,s(o;) + (1 -
5,
t,)voS(u,)
= 0.
(4.4)
We shall show that it follows from here that v,+
F(lln)
llll,,IIp_l+= h* f(. 94sJ>
in Wi.
v
in Wp!
-;i+luIP-’
(4.5)
for some h” E W,!
v+ - ;i_lo~P-‘v-
in L,,
(4.6)
(4.7)
ll~~nll~-’
(at least for a subsequence) with some j+, ;Z_ satisfying (Ef’). It follows from (4.4) that J(vn) - J(um), v, - u,) =
F(un>
( Ii lIu,llp-l 5
-
-
t
F(G)
ml/%Ap-l ~
(4.8) We can suppose v,,--, v uniformly with respect to the completely continuous imbedding l&k c C((0, A). It follows from here using (G) that the right-hand side in (4.8) tends to zero as n, m+ + x and therefore {u,} is strongly convergent in *i by (M), i.e. (4.5) holds. We have t > 0 because in the opposite case we would obtain by the limiting process from (4.4) that u is a nontrivial solution of (EP,,,,, ) and this would be the contradiction with [y,,, v,,] E A0 (see remark 2.2). Now, (4.6) follows from (4.4), (4.5) because all the other members
1099
Generalization of Fredholm alternative in (4.4)
are convergent.
Further, it follows from (G), (4.5) and the imbedding @j, C C((0, n))
that i”,$!?j is bounded in L, (for q see (G)), i.e. we can suppose
fLUn(J)_h lI4IIIp-l
inL
(4.7’)
4’
for some h E L,. We obtain from (G) that h(t) = 0 a.e. on M0 = {t E (0, n); u(r) = 0) and therefore we can write h(r) = f(t) ]u(~)/P-~ u(t) and put f+(t) = ii(t)
on M+ = {t E (0, n); u(t) 2 O},
f- (1) = a(t)
on M- = {t E (0, n); u(t) 5 0).
The uniform convergence u,--, u and (E,) or (EJ directly imply that ,Y+ and j_ satisfy (Ef=) in the sense of remark 3.1). This together with (4.7’) yields (4.7). Now, using (4.5), (4.6), (4.7) we obtain by the limiting process from (4.4) that u is a nontrivial solution of (EP,+) with x+(t) = Q+(f) + (1 - r),u,, x_(t) = Q_(t) f (1 - t)uO satisfying (El). This is the contradiction with theorem 3.2. Proof of theorem 2.1. Using remark 2.4 it is sufficient to prove the existence of the solution of the equation (AE) which is equivalent (with respect to remark 4.1) to
w + F(J_‘(w))
= g*.
(4.9)
Lemma 4.1 ensures the existence of r such that (4.2) holds, i.e. (4.10)
R(t, w) # 0, for all t E (0, l), w E Wp‘;‘, jIwjI* = R, for R sufficiently large, where R(t, w) = w - rF(J_‘(w))
- (1 - t)/%S(J-i(w))
(We have used the fact that J is (p - 1)-homeomorphism-see It is
- rg*.
remark 4.1).
deg[l-AS(J-‘);B;i,O]fO by Borsuk theorem (see e.g. [12]) b ecause S(J-‘) is odd. (I denotes the identity mapping in W;i). Using (4.10) and the homotopy invariance property of the Leray-Schauder degree, it follows from here deg[Z - F(J-‘)
- g*; BE, 0] # 0.
This implies the existence of the solution of (4.9). Remark
4.2. If [p,,, vo] E A; for some i (for the notation see remark : IV;’ --, IV;’ is defined by deg [Z,,, v,,; Bi , 0] f 0, where ZpO~yO
Z pO,yO(w)= w - ,u@s(J-l(w)+)
+ vJ(J-i(w)-).
2.2. 2.3) then
L. BOCCARDO et al.
1100
This can be proved using Borsuk theorem. the homotopy invariance property of the degree, the fact that the component Ai contains the point [A, A], where A is not an eigenvalue of (EP) and that the operator I - AS(J-‘) is odd. For details see [8]. Proof of theorem 2.2. This is the same as that of theorem 2.1 but we use lemma 4.2 instead of lemma 4.1, the homotopy
H(t, w) = w - tF(J-l(w))
- rg* - (1 - t)pOS(J-l(w)+)
+ (1 - r)VoS(J-‘(wj-)
instead of fi and remark 4.2 instead of Borsuk theorem. LEMMA 4.3. Consider the situation from theorem 2.1 and 2.2 but replace (E,), (EJ and (El) by (EL,), (EL,) and (ELI), respectively. Let A be an odd (p - 1)-homeomorphism of I@; onto Wp;’ which is (p - 1)-quasihomogeneous with respect to J. Define H, H’ as in lemmas 4.1 and 4.2 but with J replaced by A. Then the assertion of lemmas 4.1 and 4.2 in the case of theorems 2.1 and 2.2, respectively, holds. Proof. We shall consider the situation of theorem 2.2 only. The case of theorem 2.1 can be obtained from here by setting ,uo = v. = A and using theorem 3.1 instead of theorem 3.2. Suppose that our assertion is not true. Then there exist IL,,,r,, such that ]lunl\-+ + TJ, r,, E (0, l), r, ---, r, u, = u,,/]Iu,I]- u and
+ (1
- r,),uoS(u,‘)
- (1 - r,)v,S(u,).
(4.11)
It is ]~~(t)l--+ 3~and f07 un 0))
f(b un 0))
= lrln(f),P_ZU,(t) 14)lP-*%0) I141p-2
(n sufficiently large) for a.e. t E M = {t E (0, x); u(t) f 0). It follows from here fk
u&)>
IlUnllP- l
u’(t)- ~_(f)lu(t)p-2U-(t) --, 2+ (~)l~(w-~
(4.12)
a.e. on M, where f(t, s) X? = ,42yx -]sIp-*s’
Simultaneously,
it follows from (G) f(ry u, (0) ---, o
IMP- ’ a.e. on MO = (0, n)\M and therefore (4.12) holds a.e. on (0, ~7).Using Lebesgue’s theorem and (G) we obtain that the convergence in (4.12) takes place also in the Lt-norm. It follows from here (and remark 4.1) that the right-hand side in (4.11) converges in W;* to Q+(u]P-*u+
- rf-lu]P-*o-
+ (1 - t)/.&]P-‘uf
- (1 - T)Va]uIP-*u-
Generalization
of Fredholm
alternative
1101
and therefore
GunI I-+ J(u)
llUnllP -
(see definition 2.1). It is u # 0 because
according to the assumption that A is (p - l)-homeomorphism. solution of (EP,=) with x+(t) = rX+(t) + (1 - r)&,X-(t) satisfying (E;‘).
This contradicts
= Q-(t)
Hence, we have a nontrivial + (1 - r>vlJ
to theorem 3.2.
Proofofrheorem 2.3. This is the same as that of theorems 2.1, 2.2 but we use lemma 4.3 instead of lemmas 4.1, 4.2.
LEMMA 4.4. The assertion of lemma 4.3 remains true also under the assumptions (E,), (E,) and (El) if (G) is satisfied with
(;++
4=p* Proof. Arguing via contradiction
1).
we obtain again (4.11). The condition (G) implies, now,
that F(u,)
II4~-* is bounded in Lp.. We obtain from here (using completely continuous imbedding L,. c IV,-?)
F(un>
IIu,Ip-l--,h* in IV,-? for some h” E IV>‘. Analogously as in the proof of lemma 4.2 we obtain (4.7) (with q = p*) which implies that the right-hand side in (4.11) converges in WP;’ again to r~+JuIp-*u+ - r~-/u]p-*u-
+ (1 - t) /&)Iu]P-%+ - (1 - t)Y&IP-*U-.
To finish the proof we argue as in the proof of lemma 4.3. Proof of theorem 2.4. This is the same as that of theorems 2.1, 2.2 using lemma 4.4 instead
of lemmas 4.1, 4.2. Remark 4.3. For the proof of theorem 2.1 under a more restrictive assumption (G) with q = p*
and a special version of theorem 2.4 see [9]. B. PDE
LEMMA 4.5. Let the assumptions of theorem 2.5 be fulfilled and define H as in lemma 4.1 but with J, F, S from remark 2.7. Then the assertion of lemma 4.1 holds.
1102
L. BOCURDO
er al
Proof. The case (E,) is the same as in lemma 4.1 but we use theorem 3.3 instead of theorem 3.1. Consider the case (E,) and suppose that our assertion is not true. Then there exist T,, u,, such that
and
-
J(un>- tn Analogously here
as in the proof
of lemma
(1 - t,)AS(o,)
4.1 (but using (G’) instead
= 0. of (G))
(4.13) it follows
from
, u,+
F(un)
,,u.,,P_, + h*
u
in
.
in IV;’
f(.AlW IMlp-’
kVL
(4.14)
for some h” E W;‘,
-
~IuIP-~ u in L,.
(4.16)
with Ix(t) - AI 5 C~ - 6 a.e. on R. This implies
F(un> -II4w’ Hence,
AS(u,) +
(x(t) - A) lu(t)iP-%(t)w(t) dt d c; - 6.
we obtain
for n sufficiently
large and this is the contradiction.
Proof of theorem 2.5. This follows from lemma 4.5 by the same way as that of theorem lemma 4.1.
2.1 from
Proof of theorem 2.6. Combining the considerations from the proofs of lemmas 4.4. 1.5 it is easy to see that an analog of these lemmas for the situation from theorem 2.6 holds. The remainder of the proof is clear from the previous. Proof of theorem 2.7. This is the same as that of theorem on theorem 3.4 must be used. Acknowkdgemenrs-The
authors
would likr to thank
Luca Leonori
2.1 but an analog of lemma -1.1 based
for the figures.
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