A new method for the study of nonlinear eigenvalue problems

C. R. Acad. Sci. Paris, t. 328, Analyse fonctionnellelfunctional

SCrie I, p. 251-256, Analysis

A new method problems

for the study

Biagio

of nonlinear

eigenvalue

RICCERI

Department of Mathematics, E-mail: [email protected] (ReCu

1999

le 21 septemhra

Abstract.

University

1998,

accept6

of Catania,

le 1”

diremhre

95125

Catania,

Italy

1998)

In this Note we present the following result as a new tool for the study of nonlinear eigenvalue problems: Let X be a separable and reflexive real Banach space; Cp : X + R a lower semicontinuous convex functional; P : X + R a sequentially weakly lower semicontinuous functional. Assume that

and limll,~~~--+~(Q(:z) + Xq(z)) = + 3(: for all X 2 0. Then. for each r ~]a,b[, there exist X* > 0 and z* E X such that a(:~*) + X*Q(z*) = r and z:* is a local, non-absolute minimum for @(.) + X* a(.) in the weak topology of X. 0 Academic des ScienceslElsevier, Paris Une

nouvelle

propres

R&urn&

non

mkthode

pour

l’ktude

de problbmes

de valeurs

lin6aires

Dans cette Note on propose le r&dtat suivant comme tat outil nouveau pour l’e’tude de problemes de vuleurs propres non lineaires : soient X un espace de Banach reel separable et r@exif ; @ : X --+ R une ,fonctionnelle convex-e semi-continue infe’rieurement ; P : X + R une ,fonctionnelle se’quentiellement ,faiblement semicontinue infe’rieurement. On suppose que

et limii,11-+,(@(2) + X*(z)) = + CG p our tout X > 0. Alors, pour chaque ‘r ~]a. b[, il existe X’ > 0 et LC* E X tels que @(:I-*) + X*Q(zr*) = r. De plus, .r* est un minimum local, non absolu pour @( .) + X’ Q( .) p ar rapport a la topologie,faible de X. 0 Academic des ScienceslElsevier. Paris

Version frangaise abrt!ge’e Le but de cette Note est de proposer une nouvelle mkthode pour 1’Ctude de problkmes de valeurs propres non lidaires. Note prCsentCe par Gustave CHOQUET. 07644442/99/0328025

I 0 Acadtmie des Sciences/Elsevier, Paris

251

B. Ricceri Voici 1’CnoncC d’un rksultat typique qui peut &tre obtenu par notre mkthode. THI~ORI~ME 1. - Soient R C R” un ouvert born4 avec frontitre r&uliBre, et f, g : R + R deux fonctions continues, avec supEER J% f (t)dt > 0. 0 n suppose qu’il existe trois constantes positives a, q, y, avec q < 1 et y > 2, telles que :

max{lf(E)Il l.d<)l~ 5 41 + 111”) pour tout < E R, et

linl slip L’ .IX f(t)dt E-0 M’

< +co.

Alors, il existe 6 > 0 tel que, pour chaque p E [-6) S], il existe X0 > 0 de sorte que le probkme

a au moins trois solutions faibles Notre mtthode

duns W:‘*(0).

repose sur le principe gCnCra1 suivant :

THBORGME 2. - Soient X un espace topologique, I C R un intervalle et f : X x I --+ R une fonction donne’e. On suppose que : (a) pour tout ~1:E X, l’ensemble {X E I : f (x, X) 2 0} est fennt dans I et l’ensemble {A E I : f(x,A) > O} est non vide, connexe et ouvert duns I ; (b) pour tout X E I, l’ensemble {z E X : f(x, X) 5 0} est non vide, fermk et skquentiellement compact ; (c) il existe X0 E I tel que l’ensemble {z E .X : f(z, X0) 5 0} est connexe. Alors, il existe x* E X, X* E I, une suite {Xn}laE~ duns I qui converge vers A* et un voisinage U de z* tels que f(z*, X”) = 0 et f(z: X,,) > 0 p our tous n, E N, .x E U. Dow, en particulier, .z* est un minimum local pour f(.? A*). La conskquence du thCor&me 2 qui permet d’obtenir des r&hats la suivante :

analogues au thdorkme 1 est

THEORBME3. - Soient X un ensemblenon born&, fermt et convexe duns un espace de Banach r&e1 ¶ble et @flexif; 6 ~10, +w] ; @ : X -+ R une fonctionnelle semi-continue infe’rieurement et convexe ; !P : X -+ R une fonctionnelle skquentiellementfaiblement semi-continue infe’rieurement. On pose : a = sup inf (Q(z) + X@(z)), XE[O,6[x&X

b = $(a(~)

+ 6max{*(z)70}),

avec la convention +m 0 = 0, et on suppose que a < b. En outre, on suppose que lim ZES,jlXll-++CC (G(x) + m(x)) = + 30 pour tout X E [0,6[. Alors, pour chaque T E]a, b[, il existe X* ~10, S[ et x* E X tels que @(z*) + X*~(:C*) = T. De plus, z* est un minimum local, non absolu pour a(-) + X*9(.) p ar rapport ci la topologie faible relative de X.

252

A new

method

for

the

study

of nonlinear

eigenvalue

problems

1. Introduction As the title says, the aim of this Note is essentially nonlinear eigenvalue problems.

to propose a new method for the study of

We give at once the statement of a typical result which can be obtained by means of our approach. 1.1. - Let Sz 2 R” be an open bounded set, with smooth boundary, and f, g : R + R two continuous functions, with supCER Jd f(t)dt > 0. A ssume that there are three positive constants a, q, y, with q < 1 and y > 2, such that THEOREM

max{lf(t)Ij

1.9(01> I 41 + IQ9

for all < E R, and

linl sup Jd f(tw E--o kl’ Then, there exists 6 > 0 such that, for each p E [-S,6],

has at least three distinct

weak solutions

Our method is based on the following

< +cm. there exists X0 > 0 such that the problem:

in WA’“(fl). general principle:

THEOREM 1.2. - Let X be a topological space, I C R an interval and f : X x I -+ R a given function. Assume that: (a) for each 2 E X, the set {X E I : f(x, X) > 0) ts c 1ose d in I and the set {X E I : ,f(:c, X) > 0) is nonempty, connected and open in I; (b) for each X E I, the set {x E X : f(s, X) 5 0) is nonempty, closed and sequentially compact; (c) there is X0 E I such that the set {x E X : f(z, X0) < 0} is connected.

Under such assumptions, there exist IC* E X, X* E I, a sequence {Xn}71E~ in I converging to X* and a neighbourhood U of z* such that f (x*: X*) = 0 and f(x, X,) > 0 for all 11,E N, .x E U. So, in particular, x* is a local minimum for f(., X”). The consequence of Theorem I .2 which allows us to obtain

results

like Theorem 1.1 is as follows:

1.3. - Let X be an unbounded, closed, convex set in a separable and reflexive real Banach 9 : X 4 R a space E; 6 ~10, +w]; Q, : X --f R a lower semicontinuous convex functional; sequentially weakly lower semicontinuous functional. Set: THEOREM

a= with the convention

sup inf (a(X) AE[O,h[xE*x-

+ AQ(.r)):

b = f;[$@(x)

+cx, . 0 = 0, and assume that a <

Moreover,

+ 6 max{ Q(2), 0}),

b.

suppose that

253

8. Ricceri

for all A E [O,S[. Then, for each T E]a, b[, there exist X” E]O,~[ and zr* E X such that qx*> + x*xIJI(x*> = r and z* is a local, non-absolute minimum for a(.) + A*@(s) in the relative weak topology of X.

Before passing to the proofs of Theorems 1.2 and 1.3, we wish to stress that Theorem 1.1 is just the simplest example of application of Theorem 1.3: several other results can be obtained, under a variety of different assumptions. We will return to this point in forthcoming papers. 2. Preliminary

results

For the reader’s convenience, we first recall some basic notions on multifunctions. two topological spaces and F : X ---t 2’- a multifunction. For any R C Y, we set

F-(R) = {x E x : F(x) n R #

Let X, Y be

S}.

We say that F is lower (resp. upper) semicontinuous if, for every open set s1 5 Y, the set F-(52) is open (resp. closed). The graph of F (denoted by Q(F)) is the set {(z, y) E X x Y : y E F(z)}. We say that F is sequentially upper semicontinuous if for every sequence {(z,! yln)}nEN in p(F), with (zI,~} converging to some :co E X, there is a subsequence of {ylL}ILE~ converging to some yo E F(zo). It is easy to check that, when X is first countable, the sequential upper semicontinuity of F implies the upper semicontinuity of F. We now point out four propositions which will be used in the proof of Theorem 1.2. The first of them was established in [2], the second is a quite direct consequence of Theorem 1 of [l], and the remaining ones are new. PROPOSITION 2.1. - Let X be a topological space; I C R a compact interval; S a connected subset of X x I whose projection on I is equal to I; F : X * 2I a lower semicontinuous multifunction, with nonempty connected values. Then, the graph of F intersects S. PROPOSITION 2.2. - Let X? Y be two topological spaces, with X connected andJirst countable, and let F : X -+ 2’ be a lower semicontinuous and sequentially upper semicontinuous multifunction. Assume that there is some zo E X such that F(xo) IS nonempty and connected. Then, the graph of F is connected. Proof

- Consider the multifunction

G : X --+ 2*“”

defined by

G(x) = {x} x F(z) for all LI: E X. It is clear that G is lower semicontinuous and sequentially upper semicontinuous. So, since X is first countable, it is upper semicontinuous. Then, taking into account that G(so) is nonempty and connected, Theorem 1 of [I] ensures that the set lJzES G(z), that is the graph of F, cl is connected, as claimed. PROPOSITION 2.3. - Let X, Y be two topological spaces, and let F : X --+ 2” be a multt$unction, with nonempty values, such that F-(y) is open for all JJ E Y and X \ F- (Y/O) is sequentially compactfor some yo E Y. Then, for ever?, nondecreasing sequence {Yn}7LE~ of subsets of Y, with UnEN Y,, = Y, there exists n E N such that F-(Y,,) = X. Proof

- Arguing by contradiction,

assume that, for every 72 E N, there is IC,, E X such that J’(G)

254

n K,

= 0.

(1)

A new method

for the study of nonlinear

eigenvalue

problems

Fix Y E N such that ya E Y,. Thus, for each n > V, one has ya @F(zrL), that is :c, E X \ F-(90). Consequently, since this latter set is sequentially compact, the sequence {zn}nE~ has a cluster point z* E X \ F-(ya). Fix 1~* E F(z*) and choose n1 E N such that y* E Y,,. Since, by assumption, F-(y*) is a neighbourhood of z*, there exists n2 > -nt such that z:,, E F-(y*). Hence, y* E F(z,~) n YnL, contradicting (1). III PROPOSITION 2.4. - Let I C_R be an interval; X a topological space; F : I + 2” a multifunction, with sequentially compact and sequentially closed values, such that, for each :c E X, I \ F-(x) is connected and open in I. Then, F is sequentially upper semicontinuous. Proof. - Let {(Xn!z,)},E~ So, one has

be a sequence in gr(F), with {X,l}raE~ converging to some Xc E I.

x, E F(X,)

Vn

E N.

(2)

Consider the sets Nt = {n E N : X, 5 X0} and Na =

{n

E N : X, > X0).

One of them is infinite. Suppose, for instance, that N1 is so (the reasoning is similar if Nz is infinite). Let us define A = {X E I : {n E N1 : z, E F(X)}

is infinite}.

First, assume that X0 E A. Then, since F(Xo) is sequentially compact, there is a subsequenceof converging to some x0 E F(Xo), and we are done. Now, assumethat Xa $! A. So, there {xn)eN is Y E N such that 2, $ F(Xo) Vn > V. with n E Nt.

(3)

In view of (3), & > inf I. Let r = inf&N A,. We claim that T E A. Assume the contrary. Thus, there is ~1 > v such that x, 6 F(r)

Vn > ~1, with n E Nl.

(4)

Then, if n E N1 and n > ~1, from (3) and (4), we obtain that both T and X0 lie in I \ F-(x,) which is, by assumption, connected. But T 5 X, _ < Xc, and hence X,, E I \ F-(x,), against (2). Since F(r) is sequentially compact, there is an increasing sequence {r&}&N in N1 such that the Sequence{xn,}&:N converges to some x* E F(r). Now, as above, we see that, for each p l ]r, Xa[, the set {Ic E N : z,, E F(p)} is infinite. Consequently, since F(p) is sequentially closed, we have x* E F(p). But F-(x*) is closed, and so x* E F(Xo), which completes the proof. cl

3. Proof

of Theorem

1.2

Consider the multifunctions F : I + 2~~ and G : X 4 2’ defined by: F(X) = {x E x : f(X! A) 5 O} and G(z) = {A E I : f(x,x)

> O}.

Thanks to (a) and (b), we can apply Proposition 2.3 to G. Thus, if {I I.} key is a nondecreasingsequence of compact intervals, with X0 E II and lJkEN Ik = I, there is k: E N such that G-(Ik) = X. Set s = ((2, A) E x x Ik : f(z, X) 5 0)

255

B. Ricceri

as well

as

(a(z) = G(Z) n I, for all zzzE X. Observe that Q is a lower semicontinuous multifunction with nonempty connected values whose graph does not intersect S. Moreover, by (b), the projection of S on I, is equal to I,. Consequently, by Proposition 2.1, 5’ is disconnected. But S is homeomorphic to gr(qll,), and so this latter set is disconnected. From (a) and (b), we directly get that the multifunction F satisfies the assumptions of Proposition 2.4. Consequently, it is sequentially upper semicontinuous. Since (by (c)) F(Xa) is connected, we then conclude, in view of Proposition 2.2, that Fir, is not lower semicontinuous. Hence, there exist X* E Ik. a sequence {X,l}lLE~ in I, converging to A* and an open set U C X intersecting F(X*) such that F(X,,) tl U = 0 for all n E N. Of course, this means that f(z, X,) > 0 for all :r E U, 7~ E N. By (a), we infer that f(.~, X”) > 0 for all 5 E U. Finally, let .T* be any point in F(X*) n U. So. f(x*, X*) = 0, and hence IC* is a local minimum for f(., X’).

4. Proof

of Theorem

Fix T ~]a.b[,

1.3

and set f(z, A) = @(:r) + x*(5.)

- r

for all (z> A) E X x [O,S[. We wish to apply Theorem 1.2 to f, when X is endowed with the relative weak topology. Observe that b = infZcES s~~p~c[a,~l(@(~)+ X!@(Z)). So, since r < b, condition (a) is clearly satisfied. Moreover, since Cpis convex, we can satisfy condition (c) taking X0 = 0. Now, fix A E [O,S[, and consider the set L = {Z E X : a(:~) + X*(Z) < r}. Obviously, L is nonempty (since n < r) and bounded, in view of the coercivity of the functional Q(.) + A@(.). Let us show that L is weakly closed in X (and so in E, since X is weakly closed). So, let 50 E X be such that @(.x0) + A!P(zo) > T. Choose p > 0 so that Q(Z) + XQ(s) > r’ for all 5 E X, with [[z - ~011> p, Since E is separable and reflexive, the set {Z E X : 11~- :~a// 5 Q}, equipped with the relative weak topology, is metrizable. Consequently, the restriction to this set of the functional a(-) + XQ(.) is weakly lower semicontinuous. Hence, there is a neighbourhood U of x0 in the weak topology of E such that G(Z) + X9(z) > 7’ for all 3: E 17 n X, with [[:I: - n:al( < p. Such an inequality is so satisfied for all z E U n X. This shows that X \ L is weakly open in X. Then, by reflexivity and by the Eberlein-Smulyan theorem, L is also sequentially weakly compact, and therefore condition (b) is satisfied. At this point, the existence of A* and LC*as in the conclusion follows directly from Theorem 1.2. In particular, the fact that :r* is not an absolute minimum for G(.) + A**(.) follows 0 from the inequality u < 7’. Moreover, the fact that X* > 0 follows from the convexity of @.

References [I] Cristea [2] Ricceri ( 1993)

256

M., Some remarks on the Darboux property for multivalued B., Some topological mini-max theorems via an alternative 367-377.

functions, principle

Demonstratio Math. 28 (I 995) 345-352. for multifunctions, Arch. Math. (Basel)

60