On the role playedby the Fuck spectrum in the determination of critical groups in elliptic problems where the asymptotic limits may not exist

Nonlinear Analysis 49 (2002) 603 611 www.elsevier.com/locate/na On the role playedby the Fuck spectrum in the determination of critical groups in elliptic problems where the asymptotic limits may not exist Shujie Li a; ;1, Kanishka Perera b;2, Jiabao Su c;3 a Academy of Mathematics and Systems Sciences, Institute of Mathematics, Academia Sinica, Beijing 100080, People s Republic of China b Department of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA c Department of Mathematics, Capital Normal University, Beijing 100037, People s Republic of China Received22 November 1999; accepted20 June 2000 Keywords: Semilinear elliptic boundary value problems; Jumping nonlinearities; Fuck spectrum; Morse theory; Critical groups 1. Introduction and statement of the results Consider the semilinear elliptic boundary value problem u = f(x; u) in ; (1.1) u =0 on @; where is a bounded domain in R n with smooth boundary @, and f is a Caratheodory function on R such that f(x; 0) 0 and f(x; t 1 ) f(x; t 2 ) 6 C( t 1 p 2 + t 2 p 2 +1) t 1 t 2 ; t 1 ;t 2 R (1.2) Corresponding author. E-mail addresses: lisj@math03.math.ac.cn (S. Li), kperera@winnie.t.edu (K. Perera), sujb@mail. cnu.edu.cn (J. Su). 1 Supportedby NSFC andby the 973 program of National Natural Science Foundation of China. 2 Supportedby NSFC andby the Morningside Center of Mathematics at the Chinese Academy of Sciences. 3 Supportedin part by NSFC, by the Natural Science Foundation of Beijing, andby the Foundation of Beijing City s Educational Committee. 0362-546X/02/$ - see front matter c 2002 Elsevier Science Ltd. All rights reserved. PII: S0362-546X(01)00125-0

604 S. Li et al. / Nonlinear Analysis 49 (2002) 603 611 for some p (2; 2n=(n 2)). As is well-known, solutions of (1.1) are the critical points of the C 2 0 functional G(u)= u 2 2F(x; u); u H = H0 1 (); (1.3) where F(x; t) = t f(x; s)ds, andone way to obtain a non-trivial critical point is to 0 compare the critical groups of G at zero with those at innity (see, e.g., [1] or [7]). When G is C 2, it is well known that the critical groups of a non-degenerate critical point are completely determined by its Morse index. It was observed in [16 18] that even in some degenerate cases where G has a local linking near zero, C (G; 0) can be computedexplicitly via the shifting theorem. For Landesman Lazer type problems, C (G; 0) and C (G; ) were computedin [1,4,5], when G is only C 1, but assuming that the limits lim t 0 f(x; t)=t andlim t f(x; t)=t exist. In [2,3,9,10,12], C (G; 0) were computedfor problems with a jumping nonlinearity at zero, i.e., assuming only that the one-sided limits lim t 0 ± f(x; t)=t exist. Similarly, C (G; ) were computed in [11] for some resonance problems with a jumping nonlinearity at innity, i.e., when lim t + f(x; t)=t andlim t f(x; t)=t are dierent. The more general problem of computing C (G; 0) (resp. C (G; )) when f merely satises inequalities of the form a(t ) 2 + b(t + ) 2 6 f(x; t) t 6 a(t ) 2 + b(t + ) 2 ; (1.4) where t ± =max{±t; 0}, for t small (resp. large) was considered in [6]. It was assumed there that a;b; a; b satisfy certain conditions involving the eigenvalues of. The purpose of the present paper is to explore the role playedby the Fuck spectrum of in this problem. Recall that is the set of points (a; b) R 2 for which the problem u = bu + au in ; (1.5) u =0 on @ has a non-trivial solution, andconsists, at least locally, of curves emanating from the points ( l ; l ) where 1 2 denote the distinct Dirichlet eigenvalues of on. It was shown in [15] that in the square Q l =( l 1 ; l+1 ) 2, has two strictly decreasing curves C l1 ;C l2 passing through ( l ; l ) such that the regions in Q l below the lower curve C l1 andabove the upper curve C l2 are free of. Let d l denote the sum of the multiplicities of 1 ;:::; l. We shall prove Proposition 1.1. Assume that a(t ) 2 + b(t + ) 2 6 f(x; t)t 6 a(t ) 2 + b(t + ) 2 ; t 6 (1.6) for some 0. (i) If a;b l 1 and (a; b) is below C l1 ; then 0 is an isolated critical point of G and C q (G; 0) = qdl 1 Z. (ii) If (a;b) is above C l2 and a; b l+1 ; then 0 is an isolated critical point of G and C q (G; 0) = qdl Z.

Proposition 1.2. Assume that S. Li et al. / Nonlinear Analysis 49 (2002) 603 611 605 a(t ) 2 + b(t + ) 2 6 f(x; t) t 6 a(t ) 2 + b(t + ) 2 ; t M (1.7) for some M 0. (i) If a;b l 1 and (a; b) is below C l1 ; then G satises (PS) and C q (G; ) = qdl 1 Z. (ii) If (a;b) is above C l2 and a; b l+1 ; then G satises (PS) and C q (G; )= qdl Z. (iii) If a;b l 1 ; (a; b) is on C l1 ; and 2F(x; t) 6 (a )(t ) 2 +(b )(t + ) 2 ; t M (1.8) for some 0; then G satises (PS) and C dl 1 (G; ) 0. (iv) If (a;b) is on C l2 ; 2F(x; t) (a + )(t ) 2 +(b + )(t + ) 2 ; t M (1.9) for some 0; and a; b l+1 ; then G satises (PS) and C dl (G; ) 0. Remark 1.3. Using the notion of homological local linking introduced in [8] it was shown in [9] that (i) If l 1 t 2 6 2F(x; t) 6 a (t ) 2 + b(t + ) 2 ; t 6 (1.10) for some point ( a; b) onc l1 and0 is an isolatedcritical point of G, then C dl 1 (G; 0) 0. (ii) If a(t ) 2 + b(t + ) 2 6 2F(x; t) 6 l+1 t 2 ; t 6 (1.11) for some point (a;b)onc l2 and0 is an isolatedcritical point of G, then C dl (G; 0) 0. Some immediate applications are Theorem 1.4. Assume that a 0 (t ) 2 + b 0 (t + ) 2 6 f(x; t)t 6 a 0 (t ) 2 + b 0 (t + ) 2 ; t 6 ; (1.12) a(t ) 2 + b(t + ) 2 6 f(x; t)t 6 a(t ) 2 + b(t + ) 2 ; t M (1.13) for some ; M 0. Then (1:1) has a non-trivial solution in each of the following cases: (i) a 0 ;b 0 l 1 ; (a 0 ; b 0 ) is below C l1 ;a;b m 1 ; (a; b) is below C m1 or on C m1 and (1:8) holds; and l m; (ii) a 0 ;b 0 l 1 ; (a 0 ; b 0 ) is below C l1,(a;b) is above C m2 or on C m2 and (1:9) holds; a; b m+1 ; and l m +1;

606 S. Li et al. / Nonlinear Analysis 49 (2002) 603 611 (iii) (a 0 ;b 0 ) is above C l2 ; a 0 ; b 0 l+1 ;a;b m 1 ; (a; b) is below C m1 or on C m1 and (1:8) holds; and l m 1; (iv) (a 0 ;b 0 ) is above C l2, a 0 ; b 0 l+1 ; (a;b) is above C m2 or on C m2 and (1:9) holds; a; b m+1 ; and l m. Theorem 1.5. Assume that a 0 (t ) 2 + b 0 (t + ) 2 6 2F(x; t) 6 a 0 (t ) 2 + b 0 (t + ) 2 ; t 6 ; (1.14) a(t ) 2 + b(t + ) 2 6 f(x; t)t 6 a(t ) 2 + b(t + ) 2 ; t M (1.15) for some ; M 0. Then (1:1) has a non-trivial solution in each of the following cases: (i) a 0 = b 0 = l 1 ; (a 0 ; b 0 ) is on C l1 ;a;b m 1 ; (a; b) is below C m1 ; and l m; (ii) a 0 = b 0 = l 1 ; (a 0 ; b 0 ) is on C l1 ; (a;b) is above C m2 ; a; b m+1 ; and l m +1; (iii) (a 0 ;b 0 ) is on C l2 ; a 0 = b 0 = l+1 ;a;b m 1 ; (a; b) is below C m1 ; and l m 1; (iv) (a 0 ;b 0 ) is on C l2 ; a 0 = b 0 = l+1 ; (a;b) is above C m2 ; a; b m+1 ; and l m. Theorems 1.4 and1.5 extendsome existence results in [6,9 11,13]. 2. Preliminaries on From the variational point of view, solutions of (1.5) are the critical points of I(u)=I(u; a; b)= u 2 a(u ) 2 b(u + ) 2 ; u H: (2.1) Let N l denote the subspace of H spannedby the eigenfunctions corresponding to 1 ;:::; l andlet M l = Nl. It was shown in [15] that there are continuous andpositive homogeneous maps = ( ;a;b):m l 1 N l 1 ;= ( ;a;b):n l M l such that v 0 = (w); w 0 = (v) are the unique solutions of I(v 0 + w) = sup I(v + w); v N l 1 w M l 1 ; (2.2) I(v + w 0 ) = inf I(v + w); w M l v N l ; (2.3) respectively. Let M l 1 (a; b) = inf I((w)+w); (2.4) w M l 1 w =1 m l (a; b) = sup I(v + (v)); (2.5) v N l v =1 l 1 (a) = sup {b: M l 1 (a; b) 0}; (2.6) l (a) = inf {b: m l (a; b) 6 0}: (2.7)

S. Li et al. / Nonlinear Analysis 49 (2002) 603 611 607 According to [15], l 1 and l are strictly decreasing continuous functions such that l 1 ( l )= l ( l )= l, and C l1 : b = l 1 (a) and C l2 : b = l (a) are the minimal and maximal curves of in Q l, respectively. 3. Proof of Proposition 1.1 Proofs of (i) and(ii) are similar, so we consider only (i). Let G (u)=i(u; a; b) 2(1 ) F(x; u); u H; [0; 1]; (3.1) where F(x; t)=f(x; t) 1 2 (a(t ) 2 + b(t + ) 2 ). We will show that 0 is the only critical point of G t in B = {u H: u } for suciently small 0, so C q (G; 0) = C q (G 0 ; 0) = C q (G 1 ; 0) = C q (I( ; a; b); 0) = qdl 1 Z (3.2) by the homotopy invariance of critical groups andtheorem 1:1 of Perera andschechter [14]. To see that 0 is an isolatedcritical point of G, suppose, on the contrary, that there are sequences j [0; 1]; u j 0;u j 0 such that u j is a critical point of G j.we may assume that j [0; 1] andũ j =u j = u j ũ weakly in H, strongly in L 2 (), anda.e. in. Set f = @ F=@t and a j (x)= 0 otherwise; f(x; u j ) u j if 6 u j (x) 0; f(x; u j ) if 0 u j (x) 6 ; b j (x)= u j 0 otherwise: (3.3) (3.4) Since (a a)(t ) 2 ( b b)(t + ) 2 6f(x; t)t 6 0; t 6 (3.5) by (1.6), (a a) 6 a j (x) 6 0; ( b b) 6 b j (x) 6 0; (3.6) so we may assume that a j a; b j b weakly in L 2 () where (a a) 6 a(x) 6 0; ( b b) 6 b(x) 6 0: (3.7) Now for any v H, 0= (G j (u j );v) =(I (ũ j ; a; b);v) 2 u j [ ] (1 j ) (b j (x)ũ + j a j (x)ũ f(x; u j ) j )v + u j u j v ; (3.8)

608 S. Li et al. / Nonlinear Analysis 49 (2002) 603 611 andthe rst part of the right-handside converges to (I (ũ; a; b);v) (1 ) (b(x)ũ + a(x)ũ )v; (3.9) while u j by (1.2). It follows that where f(x; u j ) u j v 6 C u j p 2 v 0 (3.10) I (ũ; ã(x); b(x))=0; (3.11) ã(x)= a +(1 )a(x); b(x)= b +(1 )b(x): (3.12) Moreover, ũ 0, for otherwise 0= (G j (u j );u j ) 2 u j 2 =1 (a +(1 j )a j (x))(ũ j ) 2 +(b +(1 j )b j (x))(ũ + j ) 2 (1 j ) u j f(x; u j ) u j ũ j 1: (3.13) Write ũ =ṽ + w N l 1 M l 1. Since ã(x) a l 1 ; b(x) b l 1, I(v + w; ã(x); b(x)) is strictly concave in v N l 1, so there is a unique v 0 N l 1 such that I(v 0 + w; ã(x); b(x)) = sup I(v + w; ã(x); b(x)): (3.14) v N l 1 Setting u 0 = v 0 + w, 0=(I (ũ; ã(x); b(x)) I (u 0 ; ã(x); b(x)); ṽ v 0 ) = (ṽ v 0 ) 2 ( b(x)(ũ + u 0 + ) ã(x)(ũ u 0 ))(ũ u 0) 6 (min{a;b} l 1 ) (ṽ v 0 ) 2 ; (3.15) so ṽ = v 0. In particular, this implies that w 0, for otherwise ṽ = 0 andhence ũ =0. But then 0= 1 2 (I (ũ; ã(x); b(x)); ũ) = I(ũ; ã(x); b(x)) = sup v N l 1 I(v + w; ã(x); b(x)) I(( w; a; b)+ w; ã(x); b(x)); where was dened in Section 2

S. Li et al. / Nonlinear Analysis 49 (2002) 603 611 609 I(( w; a; b)+ w; a; b); since ã(x) 6 a; b(x) 6 b M l 1 (a; b) w 2 ; where M l 1 was dened in (2:4) 0 (3.16) by Lemma 3:12 of Schechter [15] since ( a; b) is below C l1, a contradiction. 4. Proof of Proposition 1.2 (i) Here we apply the following homotopy invariance theorem for critical groups at innity from [14] to the family of functionals dened in (3.1). Theorem 4.1. Let G ; [0; 1] be a family of C 1 functionals dened on a Hilbert space H; that satises (PS); such that G ;@G =@ are locally Lipschitz continuous. If there is an R 0 such that inf [0;1];u H\B R G (u) 0; inf G (u) ; (4.1) [0;1];u B R then C (G 0 ; ) = C (G 1 ; ): (4.2) To see that G satises (PS) andthat (4:1) holds for suciently large R, suppose that there are sequences j [0; 1]; u j ; u j such that G j (u j ) 0. We may assume that j [0; 1] andũ j = u j = u j ũ weakly in H, strongly in L 2 (), and a.e. in. Set f(x; u j ) if u j (x) M; a j (x)= u j (4.3) 0 otherwise; f(x; u j ) if u j (x) 6 M; b j (x)= u j 0 otherwise: (4.4) Then (3.6) holds by (1.7), so we may assume that a j a; b j b weakly in L 2 (); where a; b satisfy (3.7). For any v H, (G j (u j );v) =(I (ũ j ; a; b);v) 2 u j [ (1 j ) (b j (x)ũ + j a j (x)ũ j )v + u j M ] f(x; u j ) u j v (4.5)

610 S. Li et al. / Nonlinear Analysis 49 (2002) 603 611 and u j M f(x; u j ) u j v 6 C u j 1 v 0; (4.6) so it follows that ũ is a critical point of I( ; ã(x); b(x)) where ã; b are given by (3.12). If ũ = 0, then (G j (u j );u j ) 2 u j 2 =1 (a +(1 j )a j (x))(ũ j ) 2 +(b +(1 j )b j (x))(ũ + j ) 2 (1 j ) u j M f(x; u j ) u j ũ j 1; so ũ 0. Now we get a contradiction as in the proof of Proposition 1.1. Thus (4.7) C q (G; ) = C q (I( ; a; b); ) = C q (I( ; a; b); 0) = qdl 1 Z; (4.8) where the secondisomorphism holds because ( a; b). (iii) To see that G satises (PS), suppose that there is a sequence u j ; u j such that G (u j ) 0 and G(u j ) is bounded. Passing to a subsequence, ũ j =u j = u j ũ weakly in H, strongly in L 2 (), anda.e. in, andan argument similar to that in the proof of (i) shows that ũ is a critical point of I( ; ã(x); b(x)); where a 6 ã(x) 6 a; b 6 b(x) 6 b: (4.9) Moreover, (3.16) holds with the strict inequality in the last line replaced by, so ( w; a; b)+ w =ũ and I(ũ; a; b)=0: (4.10) Passing to the limit in (G (u j );u j ) 2 u j 2 1 a(ũ j ) 2 + b(ũ + j ) 2 f(x; u j ) u j M u j ũ j (4.11) gives a(ũ ) 2 + b(ũ + ) 2 1; (4.12) andcombining this with (4:10) and ũ 6 lim inf ũ j = 1 we see that ũ = 1 and hence ũ j ũ strongly in H. But then ( G(u j ) u j 2 I(ũ j ; a; b)+ ũ j 2 L C ) 2 u j 2 (4.13) by (1.8) and(4.10), a contradiction. Since min {a;b}t 2 C 6 2F(x; t) 6 a(t ) 2 + b(t + ) 2 + C (4.14)

S. Li et al. / Nonlinear Analysis 49 (2002) 603 611 611 by (1.7) and(1.8), G(v) 6 (min{a;b} l 1 ) v 2 L + C 2 as v ; v N l 1; (4.15) G((w; a; b)+w) I((w; a; b)+w; a; b) C C; w M l 1 ; (4.16) andit follows that C dl 1 (G; ) 0 as the set {(w; a; b)+w: w M l 1 } is a manifold modeled on M l 1. Proofs of (ii) and(iv) are similar to those of (i) and(iii), respectively, andare omitted. Acknowledgements The secondauthor gratefully acknowledges the support andhospitality of the Morning-side Center of Mathematics at the Chinese Academy of Sciences where this work was completed. References [1] K.C. Chang, Innite-dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Dierential Equations andtheir Applications, Vol. 6, Birkhauser Boston Inc., Boston, MA, 1993. [2] E.N. Dancer, Multiple solutions of asymptotically homogeneous problems, Ann. Math. Pure Appl. 152 (4) (1988) 63 78. [3] E.N. Dancer, Remarks on jumping nonlinearities, in: Topics in Nonlinear Analysis, Birkhauser, Basel, 1999, pp. 101 116. [4] S.J. Li, J.Q. Liu, Computations of critical groups at degenerate critical point and applications to nonlinear dierential equations with resonance, Houston J. Math. 25 (3) (1999) 563 582. [5] S.J. Li, K. Perera, Computations of critical groups in resonance problems where the nonlinearity may not be sublinear, Nonlinear Anal., to appear. [6] S.J. Li, K. Perera, J.B. Su, Computation of critical groups in elliptic boundary value problems where the asymptotic limits may not exist, Proc. Roy. Soc. Edinburgh Sect. A., to appear. [7] J. Mawhin, M. Willem, Critical Point Theory andhamiltonian Systems, AppliedMathematical Sciences, Vol. 74, Springer, New York, 1989. [8] K. Perera, Homological local linking, Abstracts Appl. Anal. 3 (1 2) (1998) 181 189. [9] K. Perera, M. Schechter, Type (II) regions between curves of the Fuck spectrum andcritical groups, Topology Methods Nonlinear Anal. 12 (2) (1998) 227 243. [10] K. Perera, M. Schechter, Computation of critical groups in Fuck resonance problems, preprint. [11] K. Perera, M. Schechter, Double resonance problems with respect to the Fuck spectrum, preprint. [12] K. Perera, M. Schechter, The Fuck spectrum andcritical groups, Proc. Amer. Math. Soc., to appear. [13] K. Perera, M. Schechter, A generalization of the Amann Zehnder theorem to nonresonance problems with jumping nonlinearities, NoDEA Nonlinear Dierential Equations Appl., 7 (4) (2000) 361 367. [14] K. Perera, M. Schechter, Solution of nonlinear equations having asymptotic limits at zero andinnity, Calc. Var. Partial Dierential Equations, to appear. [15] M. Schechter, The Fuck spectrum, Indiana Univ. Math. J. 43 (4) (1994) 1139 1157. [16] J.B. Su, Double resonant elliptic problems with the nonlinearity may grow linearly, preprint. [17] J.B. Su, Multiple solutions for semilinear elliptic resonant problems with unbounded nonlinearities, preprint. [18] J.B. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., to appear.