Lancelotti, Sergio

English

LANCELOTTI, SERGIO

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEInfinitely many solutions for one-dimensional eigenvalue problems for variational inequalities / LANCELOTTI S.. - In:ANNALI DELL'UNIVERSITÀ DI FERRARA. SEZIONE 7: SCIENZE MATEMATICHE. - ISSN 0430-3202. - 41(1995), pp.33-44.OriginalInfinitely many solutions for one-dimensional eigenvalue problems for variational inequalitiesPublisher:PublishedDOI:10.1007/BF02826006Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1401821 since:Springer VerlagINFINITELY MANY SOLUTIONS FOR ONE-DIMENSIONALEIGENVALUE PROBLEMS FOR VARIATIONAL INEQUALITIESAuthor's versionPublished in: Annali dell'Universita di Ferrara Sez. VII 41 (1995), 33-44SERGIO LANCELOTTIDipartimento di MatematicaUniversita Cattolica del Sacro CuoreVia Trieste 17, I 25121 Brescia, ItalyAbstract. A non-symmetric perturbation of a symmetric variational inequalityis considered. The existence of innitely many solutions is proved.1 IntroductionLet  : ]0; [! [0;+1] be a lower semicontinuous function and letK = fu 2 H10 (0; ) : juj   g :If p : ]0; [R! R is a Caratheodory function odd in the second variable,  > 0 andsuitable qualitative conditions are satised, it is known that the problem8>><>>:(; u) 2 RKR 0[DuD(v   u) + p(x; u)(v   u)] dx   R 0u(v   u) dx 8v 2 KR 0u2 dx = 2admits a sequence (h; uh) of solutions with h ! +1 . This has been proved, underdierent assumptions, in [4, 6, 13], even in the multi-dimensional case.This work was partially supported by a national research grant nanced by theMinistero dell'Universita e della Ricerca Scientica e Tecnologica (40% { 1992).2In [8, 12] we have shown that, if we perturb the problem in a quite general non-symmetric way, then the number of solutions of the perturbed problem goes to innity,as the perturbation tends to disappear.Here we want to treat a case in which the perturbed problem has still innitelymany solutions.In the case of equations, results of this kind have been obtained in [1, 2, 3, 14, 15,16]. The technique here is similar, from the topological point of view. To stress thisaspect, we will work with the notion of \essential value", which is purely topological.On the contrary, from the dierential point of view, the presence of the constraint Kcauses an irregularity which will be treated by the use of techniques of nonsmoothanalysis. Here we will follow the approach of [7, 9].We have restricted our attention to the one-dimensional case, because here thecritical values of the unperturbed problem have a suitable growth. A similar result,in a more restrictive form, could have been proved also in two dimensions, but ourargument does not work if the dimension is at least three.The author wishes to thank Marco Degiovanni for helpful discussions.2 Essential values of continuous functionalsIn this section we recall from [8, 12] some basic facts that will be useful in the following.Let X denote a metric space endowed with the metric d and f : X ! R a continuousfunction. If b 2 R := R [ f 1;+1g, let us setf b = fu 2 X : f(u)  bg :(2.1) DEFINITION. Let a; b 2 R with a  b. The pair  f b; fa is said to be trivial,if for every neighbourhood [0; 00] of a and [0; 00] of b 0; 00; 0; 00 2 R ; 0  0there exists a continuous map H : f0  [0; 1]! f00 such thatH(x; 0) = x 8x 2 f0 ;Hf0  f1g f00 ;Hf0  [0; 1] f00 :3(2.2) Remark. If  is a further number with  < 0 , we can suppose, without loss ofgenerality, thatH(x; t) = x on f[0; 1]. Actually, it is sucient to substituteH(x; t)with H (x; t#(x)) , where # : f0 ! [0; 1] is a continuous function with #(x) = 0 forf(x)   and #(x) = 1 for f(x)  0.(2.3) DEFINITION. A real number c is said to be an essential value of f , if for every" > 0 there exist a; b 2]c "; c+"[ with a < b such that the pair  f b; fa is not trivial.(2.4) THEOREM. Let a; b 2 R with a < b. Let us assume that f has no essential valuein ]a; b[.Then the pair f b; fais trivial.Now let us recall a notion from [7, 9].(2.5) DEFINITION. For every u 2 X let us denote by jdf j(u) the supremum of the 'sin [0;+1[ such that there exist  > 0 and a continuous map H : B(u) [0; ] ! Xwithd(H(v; t); v)  t ;f(H(v; t))  f(v)  t :The extended real number jdf j(u) is called the weak slope of f at u.If X is a Finsler manifold of class C1 and f a function of class C1, it turns outthat jdf j(u) = kdf(u)k for every u 2 X.Let us point out that the above notion has been independently introduced also in[10].(2.6) DEFINITION. An element u 2 X is said to be a critical point of f , if jdf j(u) = 0.A real number c is said to be a critical value of f , if there exists a critical point u 2 Xof f such that f(u) = c. Otherwise c is said to be a regular value of f .(2.7) DEFINITION. Let c be a real number. The function f is said to satisfy thePalais - Smale condition at level c ((PS)c for short), if every sequence (uh) in X withjdf j(uh)! 0 and f(uh)! c admits a subsequence (uhk) converging in X.(2.8) THEOREM. Let c be an essential value of f . Let us assume that X is completeand that f satises (PS)c.4Then c is a critical value of f .3 Innitely many solutions for non-symmetric variational inequalitiesLet p : ]0; [R! R be a Caratheodory function such thatp(x; s) =  p(x; s) ;s p(x; s)  0 ;8t > 0 : supjsjtjp(x; s)j 2 L1(0; ) ;let  : ]0; [! [0;+1] be a lower semicontinuous function and let  > 0 with2 <Z 0 2 dx :We start from the nonlinear eigenvalue problem8>><>>:(; u) 2 RKR 0[DuD(v   u) + p(x; u)(v   u)] dx   R 0u(v   u) dx 8v 2 KR 0u2 dx = 2(3:1)whereK =u 2 H10 (0; ) :   (x)  ~u(x)   (x) 8x 2]0; [	and ~u is the continuous representative of u. It is readily seen that (3.1) possesses asymmetry. In fact, if (; u) is a solution of (3.1), also (; u) is a solution of (3.1).We want to study a perturbation of (3.1) of the form8>>>>><>>>>>:(; u) 2 RKR 0[DuD(v   u) + (p(x; u) + q(x; u))(v   u)] dx++ < ; v   u >  R 0u(v   u) dx 8v 2 KR 0u2 dx = 2(3:2)5where q : ]0; [R ! R is a Caratheodory function and  2 H 1(0; ). We assumethatjq(x; s)j  a1(x) + bjsj#with a1 2 L1(0; ) , b 2 R and 0 < # < 3 .We want to show that problem (3.2) has innitely many solutions.Problems (3.1) and (3.2) have a variational structure. Let us setS =u 2 L2(0; ) :Zu2 dx = 2and let us dene f; g : K \ S ! R byf(u) =12Z 0jDuj2 dx +Z 0P (x; u) dx ;g(u) =12Z 0jDuj2 dx +Z 0P (x; u) dx +Z 0Q(x; u) dx+ < ; u > ;where P (x; s) =R s0p(x; t) dt ; Q(x; s) =R s0q(x; t) dt. In the following, the set K \ Swill be endowed with the H10 metric. We will denote by k  kr the norm of Lr(0; )and by k  k 1;2 the norm of H 1(0; ) .Let us recall a denition from [5].(3.3) DEFINITION. Let C be a convex subset of a Banach space X, let M be ahypersurface in X of class C1, let u 2 C \M and let (u) 2 X 0 be a unit normalvector to M at u. The sets C and M are said to be tangent at u, if we have either< (u); v   u > 0 8v 2 Cor< (u); v   u > 0 8v 2 C ;where < ;  > is the pairing between X 0 and X.The sets C and M are said to be tangent, if they are tangent at some point ofC \M .Let us recall a particular case of a characterization given in [8].(3.4) THEOREM. The following facts hold:6a) given u 2 K \ S, the sets K and S are tangent at u, if and only if8x 2]0; [ : ~u(x) 6= 0 =) j~u(x)j =  (x) ;b) the sets K and S are tangent, if and only if there exists an open subset E of ]0; [such that the function  E is continuous and belongs to H10 (0; ) \ S.(3.5) THEOREM. Let us assume that K and S are not tangent. For every h  1 letus setch = infC2 hmaxu2Cf(u) ;where  h is the family of compact subsets of K \ S of the form '(Sh 1) with ' :Sh 1 ! K \ S continuous and odd.Then the following facts hold:a) K \ S is contractible in itself;b) for every h  1 we have  h 6= ; and ch  122h2.Proof. Property a) is proved in [8]. By means of [11, Lemma VI.4.5] it is then standardto deduce that  h 6= ; for every h  1 . Finally, let f0 : H10 (0; )\S ! R be denedbyf0(u) =12Z 0jDuj2 dx :If c0h is dened as ch, with K \ S substituted by H10 (0; ) \ S and f substituted byf0 , it is well known (see [11]) that c0h =122h2 . Since K \ S  H10 (0; ) \ S andf(u)  f0(u) , it follows that ch  c0h , whence the assertion.(3.6) LEMMA. Let (dh) be a sequence of positive numbers and let  2]0; 1[ . Let usassume that there exist h0 2 N and c > 0 such that8h  h0 : 0  dh+1   dh  cdh :Then there exists h1 2 N such that8h  h1 : dh  (3c) 11  h 11  :7Proof. Let us set h = h  11  dh . In [2, Lemma (5.3)] it is shown that the sequence(h) is bounded. Let us consider two cases. If for every h  h0 we haveh+1   h <  chh 1 ;then (h) goes to 0 and the assertion follows. Otherwise there exists h1  h0 suchthath1+1   h1   ch1h 11 :Arguing as in the proof of [2, Lemma (5.3)], we deduce thath1  (3c)11  :Hence, from [2, Lemma (5.3)] it follows that for every h  h1h  maxfh1 ; [c(1  )]11  g  (3c) 11  ;and the assertion follows.(3.7) LEMMA. For every  > 0 there exists C() > 0 such thatZ 0Q(x; u) dx+ < ; u >  kDuk2 + C()whenever u 2 H10 (0; ) \ S .Proof. Let a1 = a1;1 + a1;2 and  = 1 + 2 with a1;1 2 L1(0; ) , ka1;1k1  " ,a1;2 2 L2(0; ) , 1 2 H 1(0; ) , k1k 1;2  " , 2 2 L2(0; ) .Then we have Z 0Q(x; u) dx+ < ; u >  ka1;1k1kuk1 + ka1;2k2kuk2 + b#+ 1kuk#+1#+1 + C1k1k 1;2kDuk2 + k2k2kuk2  C2"kDuk2 + (ka1;2k2 + k2k2)+ C3kuk#+322 kDuk# 122  C4"kDuk2 + C5(") :A suitable choise of " then gives the assertion.(3.8) THEOREM. Let us assume that K and S are not tangent.Then the functional g admits a sequence (dh) of essential values with dh ! +1 .8Proof. First of all, let us show that for every  > 0 there exists C() > 0 such thatfor every u 2 K \ S we havef(u)  b =) g(u)  b+ pb+ C() ; (3:9)g(u)  b =) f(u)  b+ pb+ C() : (3:10)According to the previous Lemma, there exists C1() > 0 such that8u 2 H10 (0; ) \ S :Z 0Q(x; u) dx+ < ; u >  2 kDuk2 + C1() :If f(u)  b , it follows kDuk2 p2b , henceg(u)  b+p22pb+ C1()  b+ pb+ C1() :On the other hand, if g(u)  b , we haveb  12kDuk22  2kDuk2   C1()  14kDuk22   C2() ;hencekDuk2  2pb+ C2()  2pb+ 2pC2() :It followsf(u)  b+ pb+ pC2() + C1() = b+ pb+ C3() :Therefore (3.9) and (3.10) are proved.Let  > 0 be such that 1622 < 2 and let ch 2 R be dened as in Theorem (3.5) .We claim that there exists a sequence hk ! +1 such that8k 2 N : chk +pchk +C()+qchk + pchk + C()+C() < chk+1 : (3:11)Let us argue by contradiction. Then, since ch ! +1 , there exists h0 2 N such that8h  h0 : ch+1   ch  3pch :By Lemma (3.6) there exists h1 2 N such that for every h  h1 we havech  812h2 < 122h2 . This inequality contradicts Theorem (3.5) , so that (3.11)is proved.9By Theorem (2.4), it is sucient to show that for everyM > 0 there exists a Msuch that the pair (K\S; ga) is not trivial. By contradiction, assume that there existsM > 0 such that for every a  M the pair (K \ S; ga) is trivial. Let h 2 N be suchthatch M ; ch + pch + C() + qch + pch + C() + C() < ch+1and let a 2 R be such thatch + pch + C() < a ; a+ pa+ C() < ch+1 :Let d0; ; 0; 00; d00 2 R be such that ch < d0 <  < 0 < a < 00 < d00 < ch+1 withd0 + pd0 + C()   ; 00 + p00 + C()  d00 :It follows that fd0  g and g00  fd00 . Since ch < d0, there exists ' : Sh 1 !K \ S continuous and odd with '(Sh 1)  fd0 and there exists a homotopy H :Sh 1  [0; 1] ! K \ S between ' and a constant map. Let 0 2 [0;+1[ be suchthat 0  maxfg(H(x; t)) : x 2 Sh 1; t 2 [0; 1]g . Since the pair (K \ S; ga) is trivial,there exists a continuous map  : g0  [0; 1]! K \ S such that(x; 0) = x 8x 2 g0 ;g0  f1g g00 ;g0  [0; 1] g00and(x; t) = x on g  [0; 1] :Let us dene K : Sh 1[0; 1]! fd00 by K(x; t) = (H(x; t); 1). Then K is a homotopybetween ' : Sh 1 ! fd00 and a constant map. By [11, Lemma VI.4.5] there exists : Sh ! fd00 continuous and odd. This is absurd, as d00 < ch+1.(3.12) THEOREM. Let us assume that K and S are not tangent. Then the followingfacts hold:a) the functional g is continuous;10b) for every u 2 K \ S there exist  2 R and  2 H 1(0; ) such that kk 1;2 =jdgj(u) andZ 0[DuD(v   u) + p(x; u)(v   u) + q(x; u)(v   u)] dx+ < ; v   u > Z 0u(v   u) dx+ < ; v   u > 8v 2 K ;c) the functional g satises (PS)c for every c 2 R.Proof. Up to minor modications, the proof is the same of [8].Now we can prove our main result.(3.13) THEOREM. Let us assume that K and S are not tangent.Then problem (3.2) admits a sequence of solutions (h; uh) with h ! +1 .Proof. By Theorem (3.8) the functional g admits a sequence (dh) of essential valueswith dh ! +1. By Theorem (3.12) the functional g satises the (PS)c condition forevery c 2 R. Therefore by Theorem (2.8) each essential value dh is a critical value ofg. Hence there exists a sequence (uh) of critical points of g with g(uh)! +1 and byTheorem (3.12) there exists a sequence (h) such that (h; uh) is a solution of (3.2) .We havelimhZ 0jDuhj2dx = +1and, choosing v = 0 in (3.2) ,h2 Z 0[ jDuhj2 + uhp(x; uh) ] dx +Z 0uhq(x; uh) dx+ < ; uh > :Since sp(x; s)  0 and Z 0uhq(x; uh) dx  ka1k1kuhk1 + bkuhk#+1#+1  ka1k1kuhk1 + Ckuhk#+322 kDuhk# 122 ;it follows that h ! +1 .11References[1] A. BAHRI, Topological results on a certain class of functionals and application, J.Funct. Anal. 41 (1981), 397{427.[2] A. BAHRI AND H. BERESTYCKI, A perturbation method in critical point theory andapplications, Trans. Amer. Math. Soc. 267 (1981), 1{32.[3] A. BAHRI AND P. L. LIONS, Morse index of some min max critical points. I.Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988), 1027-1037.[4] A. CANINO AND U. PERRI, Constrained problems in Banach spaces with an appli-cation to variational inequalities, Nonlinear Anal. 24 (1995), 839{856.[5] G. COBANOV, A. MARINO AND D. SCOLOZZI, Evolution equation for the eigenvalueproblem for the Laplace operator with respect to an obstacle, Rend. Accad. Naz.Sci. XL Mem. Mat. 14 (1990), 139{162.[6] G. COBANOV, A. MARINO AND D. SCOLOZZI, Multiplicity of eigenvalues for theLaplace operator with respect to an obstacle, and nontangency condition, NonlinearAnal. 15 (1990), 199{215.[7] J. N. CORVELLEC, M. DEGIOVANNI AND M. MARZOCCHI, Deformation propertiesfor continuous functionals and critical point theory, Topol. Methods NonlinearAnal. 1 (1993), 151{171.[8] M. DEGIOVANNI AND S. LANCELOTTI, Perturbations of even nonsmooth functionals,Dierential Integral Equations (8) 5 (1995), 981{992.[9] M. DEGIOVANNI AND M. MARZOCCHI, A critical point theory for nonsmooth func-tionals, Ann. Mat. Pura Appl. (4) 167 (1994), 73{100.[10] G. KATRIEL, Mountain pass theorems and global homeomorphism theorems, Ann.Inst. H. Poincare Anal. Non Lineaire 11 (1994), 189{209.[11] M. A. KRASNOSELSKII, Topological methods in the theory of nonlinear integralequations, Gosudarstv. Izdat. Tehn. { Teor. Lit., Moscow, 1956. The MacmillanCo., New York, 1964.[12] S. LANCELOTTI, Perturbations of symmetric constraints in eigenvalue problems forvariational inequalities, Nonlinear Anal. 27 (1996), 633{644.12[13] A. LEACI AND D. SCOLOZZI, Esistenza e molteplicita per gli autovalori non linearidell'operatore    g rispetto a due ostacoli, Ann. Univ. Ferrara Sez. VII (N.S.)35 (1989), 71{98.[14] A. MARINO AND G. PRODI, Metodi perturbativi nella teoria di Morse, Boll. Un.Mat. Ital. (4) 11 (1975), no. 3, suppl., 1{32.[15] P. H. RABINOWITZ, Multiple critical points of perturbed symmetric functionals,Trans. Amer. Math. Soc. 272 (1982), 753{769.[16] M. STRUWE, Innitely many critical points for functionals which are not evenand applications to superlinear boundary value problems, Manuscripta Math. 32(1980), 335{364.

Infinitely many solutions for one-dimensional eigenvalue problems for variational inequalities

PORTO Publications Open Repository TOrino

INFINITELY MANY SOLUTIONS FOR ONE-DIMENSIONAL
EIGENVALUE PROBLEMS FOR VARIATIONAL INEQUALITIES
Author's version
Published in: Annali dell'Universita di Ferrara Sez. VII 41 (1995), 33-44
SERGIO LANCELOTTI
Dipartimento di Matematica
Universita Cattolica del Sacro Cuore
Via Trieste 17, I 25121 Brescia, Italy
Abstract. A non-symmetric perturbation of a symmetric variational inequality
is considered. The existence of innitely many solutions is proved.
1 Introduction
Let  : ]0; [! [0;+1] be a lower semicontinuous function and let
K = fu 2 H10 (0; ) : juj   g :
If p : ]0; [R! R is a Caratheodory function odd in the second variable,  > 0 and
suitable qualitative conditions are satised, it is known that the problem8>><>>:
(; u) 2 RKR 
0
[DuD(v   u) + p(x; u)(v   u)] dx   R 
0
u(v   u) dx 8v 2 KR 
0
u2 dx = 2
admits a sequence (h; uh) of solutions with h ! +1 . This has been proved, under
dierent assumptions, in [4, 6, 13], even in the multi-dimensional case.
This work was partially supported by a national research grant nanced by the
Ministero dell'Universita e della Ricerca Scientica e Tecnologica (40% { 1992).
2In [8, 12] we have shown that, if we perturb the problem in a quite general non-
symmetric way, then the number of solutions of the perturbed problem goes to innity,
as the perturbation tends to disappear.
Here we want to treat a case in which the perturbed problem has still innitely
many solutions.
In the case of equations, results of this kind have been obtained in [1, 2, 3, 14, 15,
16]. The technique here is similar, from the topological point of view. To stress this
aspect, we will work with the notion of \essential value", which is purely topological.
On the contrary, from the dierential point of view, the presence of the constraint K
causes an irregularity which will be treated by the use of techniques of nonsmooth
analysis. Here we will follow the approach of [7, 9].
We have restricted our attention to the one-dimensional case, because here the
critical values of the unperturbed problem have a suitable growth. A similar result,
in a more restrictive form, could have been proved also in two dimensions, but our
argument does not work if the dimension is at least three.
The author wishes to thank Marco Degiovanni for helpful discussions.
2 Essential values of continuous functionals
In this section we recall from [8, 12] some basic facts that will be useful in the following.
Let X denote a metric space endowed with the metric d and f : X ! R a continuous
function. If b 2 R := R [ f 1;+1g, let us set
f b = fu 2 X : f(u)  bg :
(2.1) DEFINITION. Let a; b 2 R with a  b. The pair  f b; fa is said to be trivial,
if for every neighbourhood [0; 00] of a and [0; 00] of b
 
0; 00; 0; 00 2 R ; 0  0
there exists a continuous map H : f0  [0; 1]! f00 such that
H(x; 0) = x 8x 2 f0 ;
H

f
0  f1g

 f00 ;
H

f
0  [0; 1]

 f00 :
3(2.2) Remark. If  is a further number with  < 0 , we can suppose, without loss of
generality, thatH(x; t) = x on f[0; 1]. Actually, it is sucient to substituteH(x; t)
with H (x; t#(x)) , where # : f0 ! [0; 1] is a continuous function with #(x) = 0 for
f(x)   and #(x) = 1 for f(x)  0.
(2.3) DEFINITION. A real number c is said to be an essential value of f , if for every
" > 0 there exist a; b 2]c "; c+"[ with a < b such that the pair  f b; fa is not trivial.
(2.4) THEOREM. Let a; b 2 R with a < b. Let us assume that f has no essential value
in ]a; b[.
Then the pair
 
f b; fa

is trivial.
Now let us recall a notion from [7, 9].
(2.5) DEFINITION. For every u 2 X let us denote by jdf j(u) the supremum of the 's
in [0;+1[ such that there exist  > 0 and a continuous map H : B(u) [0; ] ! X
with
d(H(v; t); v)  t ;
f(H(v; t))  f(v)  t :
The extended real number jdf j(u) is called the weak slope of f at u.
If X is a Finsler manifold of class C1 and f a function of class C1, it turns out
that jdf j(u) = kdf(u)k for every u 2 X.
Let us point out that the above notion has been independently introduced also in
[10].
(2.6) DEFINITION. An element u 2 X is said to be a critical point of f , if jdf j(u) = 0.
A real number c is said to be a critical value of f , if there exists a critical point u 2 X
of f such that f(u) = c. Otherwise c is said to be a regular value of f .
(2.7) DEFINITION. Let c be a real number. The function f is said to satisfy the
Palais - Smale condition at level c ((PS)c for short), if every sequence (uh) in X with
jdf j(uh)! 0 and f(uh)! c admits a subsequence (uhk) converging in X.
(2.8) THEOREM. Let c be an essential value of f . Let us assume that X is complete
and that f satises (PS)c.
4Then c is a critical value of f .
3 Innitely many solutions for non-symmetric variational inequalities
Let p : ]0; [R! R be a Caratheodory function such that
p(x; s) =  p(x; s) ;
s p(x; s)  0 ;
8t > 0 : sup
jsjt
jp(x; s)j 2 L1(0; ) ;
let  : ]0; [! [0;+1] be a lower semicontinuous function and let  > 0 with
2 <
Z 
0
 2 dx :
We start from the nonlinear eigenvalue problem8>><>>:
(; u) 2 RKR 
0
[DuD(v   u) + p(x; u)(v   u)] dx   R 
0
u(v   u) dx 8v 2 KR 
0
u2 dx = 2
(3:1)
where
K =

u 2 H10 (0; ) :   (x)  ~u(x)   (x) 8x 2]0; [
	
and ~u is the continuous representative of u. It is readily seen that (3.1) possesses a
symmetry. In fact, if (; u) is a solution of (3.1), also (; u) is a solution of (3.1).
We want to study a perturbation of (3.1) of the form8>>>>><>>>>>:
(; u) 2 RKR 
0
[DuD(v   u) + (p(x; u) + q(x; u))(v   u)] dx+
+ < ; v   u >  R 
0
u(v   u) dx 8v 2 KR 
0
u2 dx = 2
(3:2)
5where q : ]0; [R ! R is a Caratheodory function and  2 H 1(0; ). We assume
that
jq(x; s)j  a1(x) + bjsj#
with a1 2 L1(0; ) , b 2 R and 0 < # < 3 .
We want to show that problem (3.2) has innitely many solutions.
Problems (3.1) and (3.2) have a variational structure. Let us set
S =

u 2 L2(0; ) :
Z


u2 dx = 2

and let us dene f; g : K \ S ! R by
f(u) =
1
2
Z 
0
jDuj2 dx +
Z 
0
P (x; u) dx ;
g(u) =
1
2
Z 
0
jDuj2 dx +
Z 
0
P (x; u) dx +
Z 
0
Q(x; u) dx+ < ; u > ;
where P (x; s) =
R s
0
p(x; t) dt ; Q(x; s) =
R s
0
q(x; t) dt. In the following, the set K \ S
will be endowed with the H10 metric. We will denote by k  kr the norm of Lr(0; )
and by k  k 1;2 the norm of H 1(0; ) .
Let us recall a denition from [5].
(3.3) DEFINITION. Let C be a convex subset of a Banach space X, let M be a
hypersurface in X of class C1, let u 2 C \M and let (u) 2 X 0 be a unit normal
vector to M at u. The sets C and M are said to be tangent at u, if we have either
< (u); v   u > 0 8v 2 C
or
< (u); v   u > 0 8v 2 C ;
where < ;  > is the pairing between X 0 and X.
The sets C and M are said to be tangent, if they are tangent at some point of
C \M .
Let us recall a particular case of a characterization given in [8].
(3.4) THEOREM. The following facts hold:
6a) given u 2 K \ S, the sets K and S are tangent at u, if and only if
8x 2]0; [ : ~u(x) 6= 0 =) j~u(x)j =  (x) ;
b) the sets K and S are tangent, if and only if there exists an open subset E of ]0; [
such that the function  E is continuous and belongs to H
1
0 (0; ) \ S.
(3.5) THEOREM. Let us assume that K and S are not tangent. For every h  1 let
us set
ch = inf
C2 h
max
u2C
f(u) ;
where  h is the family of compact subsets of K \ S of the form '(Sh 1) with ' :
Sh 1 ! K \ S continuous and odd.
Then the following facts hold:
a) K \ S is contractible in itself;
b) for every h  1 we have  h 6= ; and ch  122h2.
Proof. Property a) is proved in [8]. By means of [11, Lemma VI.4.5] it is then standard
to deduce that  h 6= ; for every h  1 . Finally, let f0 : H10 (0; )\S ! R be dened
by
f0(u) =
1
2
Z 
0
jDuj2 dx :
If c0h is dened as ch, with K \ S substituted by H10 (0; ) \ S and f substituted by
f0 , it is well known (see [11]) that c
0
h =
1
2
2h2 . Since K \ S  H10 (0; ) \ S and
f(u)  f0(u) , it follows that ch  c0h , whence the assertion.
(3.6) LEMMA. Let (dh) be a sequence of positive numbers and let  2]0; 1[ . Let us
assume that there exist h0 2 N and c > 0 such that
8h  h0 : 0  dh+1   dh  cdh :
Then there exists h1 2 N such that
8h  h1 : dh  (3c) 11  h 11  :
7Proof. Let us set h = h
  11  dh . In [2, Lemma (5.3)] it is shown that the sequence
(h) is bounded. Let us consider two cases. If for every h  h0 we have
h+1   h <  chh 1 ;
then (h) goes to 0 and the assertion follows. Otherwise there exists h1  h0 such
that
h1+1   h1   ch1h 11 :
Arguing as in the proof of [2, Lemma (5.3)], we deduce that
h1  (3c)
1
1  :
Hence, from [2, Lemma (5.3)] it follows that for every h  h1
h  maxfh1 ; [c(1  )]
1
1  g  (3c) 11  ;
and the assertion follows.
(3.7) LEMMA. For every  > 0 there exists C() > 0 such thatZ 
0
Q(x; u) dx+ < ; u >
  kDuk2 + C()
whenever u 2 H10 (0; ) \ S .
Proof. Let a1 = a1;1 + a1;2 and  = 1 + 2 with a1;1 2 L1(0; ) , ka1;1k1  " ,
a1;2 2 L2(0; ) , 1 2 H 1(0; ) , k1k 1;2  " , 2 2 L2(0; ) .
Then we have Z 
0
Q(x; u) dx+ < ; u >
 
 ka1;1k1kuk1 + ka1;2k2kuk2 + b
#+ 1
kuk#+1#+1 + C1k1k 1;2kDuk2 + k2k2kuk2 
 C2"kDuk2 + (ka1;2k2 + k2k2)+ C3kuk
#+3
2
2 kDuk
# 1
2
2  C4"kDuk2 + C5(") :
A suitable choise of " then gives the assertion.
(3.8) THEOREM. Let us assume that K and S are not tangent.
Then the functional g admits a sequence (dh) of essential values with dh ! +1 .
8Proof. First of all, let us show that for every  > 0 there exists C() > 0 such that
for every u 2 K \ S we have
f(u)  b =) g(u)  b+ 
p
b+ C() ; (3:9)
g(u)  b =) f(u)  b+ 
p
b+ C() : (3:10)
According to the previous Lemma, there exists C1() > 0 such that
8u 2 H10 (0; ) \ S :
Z 
0
Q(x; u) dx+ < ; u >
  2 kDuk2 + C1() :
If f(u)  b , it follows kDuk2 
p
2b , hence
g(u)  b+
p
2
2

p
b+ C1()  b+ 
p
b+ C1() :
On the other hand, if g(u)  b , we have
b  1
2
kDuk22  

2
kDuk2   C1()  1
4
kDuk22   C2() ;
hence
kDuk2  2
p
b+ C2()  2
p
b+ 2
p
C2() :
It follows
f(u)  b+ 
p
b+ 
p
C2() + C1() = b+ 
p
b+ C3() :
Therefore (3.9) and (3.10) are proved.
Let  > 0 be such that 1622 < 2 and let ch 2 R be dened as in Theorem (3.5) .
We claim that there exists a sequence hk ! +1 such that
8k 2 N : chk +
p
chk +C()+
q
chk + 
p
chk + C()+C() < chk+1 : (3:11)
Let us argue by contradiction. Then, since ch ! +1 , there exists h0 2 N such that
8h  h0 : ch+1   ch  3pch :
By Lemma (3.6) there exists h1 2 N such that for every h  h1 we have
ch  812h2 < 122h2 . This inequality contradicts Theorem (3.5) , so that (3.11)
is proved.
9By Theorem (2.4), it is sucient to show that for everyM > 0 there exists a M
such that the pair (K\S; ga) is not trivial. By contradiction, assume that there exists
M > 0 such that for every a  M the pair (K \ S; ga) is trivial. Let h 2 N be such
that
ch M ; ch + pch + C() + 
q
ch + 
p
ch + C() + C() < ch+1
and let a 2 R be such that
ch + 
p
ch + C() < a ; a+ 
p
a+ C() < ch+1 :
Let d0; ; 0; 00; d00 2 R be such that ch < d0 <  < 0 < a < 00 < d00 < ch+1 with
d0 + 
p
d0 + C()   ; 00 + 
p
00 + C()  d00 :
It follows that fd
0  g and g00  fd00 . Since ch < d0, there exists ' : Sh 1 !
K \ S continuous and odd with '(Sh 1)  fd0 and there exists a homotopy H :
Sh 1  [0; 1] ! K \ S between ' and a constant map. Let 0 2 [0;+1[ be such
that 0  maxfg(H(x; t)) : x 2 Sh 1; t 2 [0; 1]g . Since the pair (K \ S; ga) is trivial,
there exists a continuous map  : g
0  [0; 1]! K \ S such that
(x; 0) = x 8x 2 g0 ;


g
0  f1g

 g00 ;


g
0  [0; 1]

 g00
and
(x; t) = x on g  [0; 1] :
Let us dene K : Sh 1[0; 1]! fd00 by K(x; t) = (H(x; t); 1). Then K is a homotopy
between ' : Sh 1 ! fd00 and a constant map. By [11, Lemma VI.4.5] there exists
 : Sh ! fd00 continuous and odd. This is absurd, as d00 < ch+1.
(3.12) THEOREM. Let us assume that K and S are not tangent. Then the following
facts hold:
a) the functional g is continuous;
10
b) for every u 2 K \ S there exist  2 R and  2 H 1(0; ) such that kk 1;2 =
jdgj(u) andZ 
0
[DuD(v   u) + p(x; u)(v   u) + q(x; u)(v   u)] dx+ < ; v   u >
 
Z 
0
u(v   u) dx+ < ; v   u > 8v 2 K ;
c) the functional g satises (PS)c for every c 2 R.
Proof. Up to minor modications, the proof is the same of [8].
Now we can prove our main result.
(3.13) THEOREM. Let us assume that K and S are not tangent.
Then problem (3.2) admits a sequence of solutions (h; uh) with h ! +1 .
Proof. By Theorem (3.8) the functional g admits a sequence (dh) of essential values
with dh ! +1. By Theorem (3.12) the functional g satises the (PS)c condition for
every c 2 R. Therefore by Theorem (2.8) each essential value dh is a critical value of
g. Hence there exists a sequence (uh) of critical points of g with g(uh)! +1 and by
Theorem (3.12) there exists a sequence (h) such that (h; uh) is a solution of (3.2) .
We have
lim
h
Z 
0
jDuhj2dx = +1
and, choosing v = 0 in (3.2) ,
h
2 
Z 
0
[ jDuhj2 + uhp(x; uh) ] dx +
Z 
0
uhq(x; uh) dx+ < ; uh > :
Since sp(x; s)  0 and Z 
0
uhq(x; uh) dx
  ka1k1kuhk1 + bkuhk#+1#+1 
 ka1k1kuhk1 + Ckuhk
#+3
2
2 kDuhk
# 1
2
2 ;
it follows that h ! +1 .
11
References
[1] A. BAHRI, Topological results on a certain class of functionals and application, J.
Funct. Anal. 41 (1981), 397{427.
[2] A. BAHRI AND H. BERESTYCKI, A perturbation method in critical point theory and
applications, Trans. Amer. Math. Soc. 267 (1981), 1{32.
[3] A. BAHRI AND P. L. LIONS, Morse index of some min max critical points. I.
Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988), 1027-
1037.
[4] A. CANINO AND U. PERRI, Constrained problems in Banach spaces with an appli-
cation to variational inequalities, Nonlinear Anal. 24 (1995), 839{856.
[5] G. COBANOV, A. MARINO AND D. SCOLOZZI, Evolution equation for the eigenvalue
problem for the Laplace operator with respect to an obstacle, Rend. Accad. Naz.
Sci. XL Mem. Mat. 14 (1990), 139{162.
[6] G. COBANOV, A. MARINO AND D. SCOLOZZI, Multiplicity of eigenvalues for the
Laplace operator with respect to an obstacle, and nontangency condition, Nonlinear
Anal. 15 (1990), 199{215.
[7] J. N. CORVELLEC, M. DEGIOVANNI AND M. MARZOCCHI, Deformation properties
for continuous functionals and critical point theory, Topol. Methods Nonlinear
Anal. 1 (1993), 151{171.
[8] M. DEGIOVANNI AND S. LANCELOTTI, Perturbations of even nonsmooth functionals,
Dierential Integral Equations (8) 5 (1995), 981{992.
[9] M. DEGIOVANNI AND M. MARZOCCHI, A critical point theory for nonsmooth func-
tionals, Ann. Mat. Pura Appl. (4) 167 (1994), 73{100.
[10] G. KATRIEL, Mountain pass theorems and global homeomorphism theorems, Ann.
Inst. H. Poincare Anal. Non Lineaire 11 (1994), 189{209.
[11] M. A. KRASNOSELSKII, Topological methods in the theory of nonlinear integral
equations, Gosudarstv. Izdat. Tehn. { Teor. Lit., Moscow, 1956. The Macmillan
Co., New York, 1964.
[12] S. LANCELOTTI, Perturbations of symmetric constraints in eigenvalue problems for
variational inequalities, Nonlinear Anal. 27 (1996), 633{644.
12
[13] A. LEACI AND D. SCOLOZZI, Esistenza e molteplicita per gli autovalori non lineari
dell'operatore    g rispetto a due ostacoli, Ann. Univ. Ferrara Sez. VII (N.S.)
35 (1989), 71{98.
[14] A. MARINO AND G. PRODI, Metodi perturbativi nella teoria di Morse, Boll. Un.
Mat. Ital. (4) 11 (1975), no. 3, suppl., 1{32.
[15] P. H. RABINOWITZ, Multiple critical points of perturbed symmetric functionals,
Trans. Amer. Math. Soc. 272 (1982), 753{769.
[16] M. STRUWE, Innitely many critical points for functionals which are not even
and applications to superlinear boundary value problems, Manuscripta Math. 32
(1980), 335{364.


A critical point theory for nonsmooth functionals,

A perturbation method in critical point theory and applications,

Constrained problems in Banach spaces with an application to variational inequalities,

Deformation properties for continuous functionals and critical point theory,

Esistenza e molteplicit a per gli autovalori non lineari dell'operatore −∆−g rispetto a due ostacoli,

Evolution equation for the eigenvalue problem for the Laplace operator with respect to an obstacle,

In many critical points for functionals which are not even and applications to superlinear boundary value problems,

Metodi perturbativi nella teoria di Morse,

Morse index of some min−max critical points. I. Application to multiplicity results,

Mountain pass theorems and global homeomorphism theorems,

Multiple critical points of perturbed symmetric functionals,

Multiplicity of eigenvalues for the Laplace operator with respect to an obstacle, and nontangency condition,

Perturbations of even nonsmooth functionals,

Perturbations of symmetric constraints in eigenvalue problems for variational inequalities,

Topological methods in the theory of nonlinear integral equations,

Topological results on a certain class of functionals and application,

https://iris.polito.it/retrieve/handle/11583/1401821/45138/Infinitely%20many%20solutions%20etc.%20%28Author%20version%29.pdf

Infinitely many solutions for one-dimensional eigenvalue problems for variational inequalities

Abstract

Similar works

Full text

Available Versions

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

PORTO Publications Open Repository TOrino