Multiple non-trivial solutions of the Neumann problem for p-laplacian systems

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1 Complex Varables and Ellptc Equatons Vol. 55, Nos. 5 6, May June 2010, Multple non-trval solutons of the Neumann problem for p-laplacan systems Sad El Manoun a and Kanshka Perera b * a Department of Mathematcs, Al-Imam Unversty, Ryadh, Saud Araba; b Department of Mathematcal Scences, Florda Insttute of Technology, Melbourne, FL, USA Communcated by R.P. Glbert (Fnal verson receved 19 September 2008) We obtan multple non-trval solutons of the Neumann problem for p-laplacan systems usng Morse theory. Keywords: p-laplacan systems; Neumann problem; multple non-trval solutons; Morse theory; non-lnear egenvalue problems; ndefnte weghts; cohomologcal ndex; non-trval crtcal groups AMS Subject Classfcatons: prmary 35J50; secondary 47J10; 58E05 1. Introducton Consder the problem 8 < p u þ a ðxþju j p 2 u ¼ F u ðx, ¼ 0 n ¼ 1,..., m where s a bounded doman n R n wth C 1 each p 2 (1, 1), p u ¼ dv(jru j p 2 ru ) s the p -Laplacan of u, a 2 L 1 () wth ess nf a 40, F 2 C 1 ( R m ) wth F(x,0)0, u ¼ (u 1,..., u m ), 2 R s a parameter s the exteror normal dervatve We assume that the non-lneartes F u satsfy the subcrtcal growth condtons:! jf u ðx, uþj C Xm ju j j rj 1 þ 1 8ðx, uþ 2 R m ð1:2þ j¼1 ð1:1þ for some r j 2ð1, 1 þ p j =ð p Þ0 Þ, where ( np p, p 5 n ¼ n p 1, p n ð1:3þ *Correspondng author. Emal: kperera@ft.edu ISSN prnt/issn onlne ß 2010 Taylor & Francs DOI: /

2 574 S. El Manoun and K. Perera s the crtcal exponent for the Sobolev mbeddng W 1,p (),! L r () and ð p Þ0 ¼ p =ð p 1Þ s the Ho lder conjugate of p and C40. Here W 1,p () s the usual Sobolev space wth the norm 1=p ku k ¼ jru j p þ a ðxþju j p : ð1:4þ We recall that a weak soluton of the system (1.1) s any u 2 W ¼ W 1,p 1 () W 1,p m () such that jru j p 2 ru rv þ a ðxþju j p 2 u v ¼ F u ðx, uþv 8v 2 W 1, p ðþ, ¼ 1,..., m: ð1:5þ They concde wth the crtcal ponts of the C 1 -functonal X m 1 ðuþ ¼ jru j p þ a ðxþju j p Fðx, uþ, p u 2 W: ð1:6þ The purpose of ths artcle s to obtan multple non-trval weak solutons usng Morse theory. To the best of our knowledge, the Neumann problem for p-laplacan systems has not been consdered n the lterature. For the scalar case, see, for example, Anello and Cordaro [1,2], Bonanno [3], Bonanno and Candto [4], Rccer [5] and the references theren. We assume that u ¼ 0 s a soluton of (1.1) and the behavour of F near zero s gven by where Fðx, uþ ¼Jðx, uþþgðx, uþ ð1:7þ Jðx, uþ ¼VðxÞju 1 j r 1 ju m j r m ð1:8þ wth r 2 (1, p ) and r 1 /p 1 þþr m /p m ¼ 1, V 2 L 1 () s a (possbly ndefnte) weght functon and G s a hgher-order term: jgðx, uþj C Xm ju j s 8ðx, uþ 2 R m ð1:9þ for some s 2ðp, p Þ. The assocated egenvalue problem 8 < p u þ a ðxþju j p 2 u ¼ J u ðx, ¼ 0 ¼ 1,..., m ð1:10þ : has non-decreasng (resp. non-ncreasng) and unbounded sequences of postve (resp. negatve) varatonal egenvalues ð k Þ when V40 (resp.50) on sets of postve measure (Secton 2). When V 0 (resp. 0) a.e. we set 1 ¼1for convenence. We also assume that F s ( p 1,..., p m )-sublnear:! jfðx, uþj C Xm ju j t þ 1 8ðx, uþ 2 R m ð1:11þ for some t 2 (0, p ).

3 Our man result s as follows: Complex Varables and Ellptc Equatons 575 THEOREM 1.1 Assume (1.2), (1.7) (1.9) and (1.11). If kþ1 5 5 k or þ k 5 5 þ kþ1 for some k 1, then (1.1) has at least two non-trval solutons. Let be a C 1 -functonal defned on a real Banach space W. We recall that n Morse theory the local behavour of near an solated crtcal pont u 0 s descrbed by the sequence of crtcal groups C q ð, u 0 Þ¼H q ð c \ U, c \ U nfu 0 gþ, q 0 ð1:12þ where c ¼ (u 0 ) s the correspondng crtcal value, c s the sublevel set {u 2 W: (u) c}, U s a neghbourhood of u 0 contanng no other crtcal ponts and H denotes Alexander Spaner cohomology wth 2 -coeffcents (see, for example, [6]). We also recall that satsfes the Palas Smale (PS) compactness condton f every sequence (u j ) W such that ðu j Þ s bounded, 0 ðu j Þ!0, ð1:13þ called a PS sequence, has a convergent subsequence. We wll prove Theorem 1.1 usng the followng three crtcal ponts theorem of Lu [7]. PROPOSITION 1.2 Assume that s bounded from below and satsfes PS. If C k (,0)6¼ 0 for some k 1, then has at least two non-trval crtcal ponts. We wll show that C k (,0)6¼ 0 under the hypotheses of Theorem 1.1 usng some recent results of Perera et al. [8] on non-trval crtcal groups n non-lnear egenvalue problems and related perturbed systems, whch we wll recall n the next secton. 2. Prelmnares For ¼ 1,..., m, let (W, kk ) be a real reflexve Banach space wth the dual ðw, kk Þ and the dualty parng h,. Then ther product W ¼ W 1 W m ¼ u ¼ðu 1,..., u m Þ: u 2 W s also a reflexve Banach space wth the norm and has the dual wth the parng and the dual norm kuk ¼ Xm ð2:1þ! 1=2 ku k 2 ð2:2þ W ¼ W 1 W m ¼ L ¼ðL 1,..., L m Þ: L 2 W, ð2:3þ hl, u ¼ Xm hl, u ð2:4þ klk ¼ Xm! 1=2 kl k 2 : ð2:5þ

4 576 S. El Manoun and K. Perera Consder the system of operator equatons A p u ¼ F 0 ðuþ ð2:6þ n W*, where p ¼ ( p 1,..., p m ) wth each p 2 (1, 1), A p u ¼ðA p1 u 1,..., A pm u m Þ, ð2:7þ A p 2 CðW, W Þ s (A 1 ) (p 1)-homogeneous and odd: A p ðu Þ¼jj p 2 A p u 8u 2 W, 2 R, ð2:8þ (A 2 ) unformly postve: 9c 40 such that A p u, u c ku k p 8u 2 W, ð2:9þ (A 3 ) a potental operator: there s a functonal I p 2 C 1 (W, R), called a potental for A p, such that (A 4 ) A p s of type (S ): for any sequence (u j ) W, I 0 p ðu Þ¼A p u 8u 2 W, ð2:10þ u j * u, A p u j, u j u! 0 ¼) u j! u, ð2:11þ and F 2 C 1 (W, R) wth F 0 ¼ (F u1,..., F um ):W! W* compact and F(0) ¼ 0. PROPOSITION 2.1 (Proposton of [8]) If each W s unformly convex and A p u, v r ku k p 1 kv k, A p u, v ¼ r ku k p 8u, v 2 W ð2:12þ for some r 40, then (A 4 ) holds. By Proposton of [8], A p s also a potental operator and the potental I p of A p satsfyng I p (0) ¼ 0 s gven by I p ðuþ ¼ Xm 1 A p u, u p : ð2:13þ Now the solutons of the system (2.6) concde wth the crtcal ponts of the C 1 -functonal PROPOSITION 2.2 (Lemma of [8]) subsequence. ðuþ ¼I p ðuþ FðuÞ, u 2 W: ð2:14þ Every bounded PS sequence of has a convergent Unlke n the scalar case, here the functonal I p s not homogeneous except when p 1 ¼¼p m. However, I p stll has the followng weaker property. Defne a contnuous flow on W by Then R W! W, ð, uþ 7! u :¼ðjj 1=p 1 1 u 1,..., jj 1=p m 1 u m Þ: ð2:15þ I p ðu Þ¼jjI p ðuþ ð2:16þ

5 Complex Varables and Ellptc Equatons 577 by (A 1 ). Ths suggests that the approprate egenvalue problems to study for the operator A p are of the form A p u ¼ J 0 ðuþ ð2:17þ where the functonal J 2 C 1 (W, R) satsfes Jðu Þ¼jj JðuÞ 8 2 R, u 2 W ð2:18þ and J 0 s compact. Takng ¼ 0 shows that J(0) ¼ 0, and takng ¼ 1 shows that J s even, so J 0 s odd, n partcular, J 0 (0) ¼ 0. Moreover, f u s an egenvector assocated wth, then so s u for any 6¼ 0 (see Proposton of [8]). Let M¼ u 2 W: I p ðuþ ¼1, M ¼ u 2M: JðuÞ? 0 : ð2:19þ Then MWn{0} s a bounded complete symmetrc C 1 -Fnsler manfold radally homeomorphc to the unt sphere n W, M are symmetrc open submanfolds of M, and the postve (resp. negatve) egenvalues of (2.17) concde wth the crtcal values of the even functonals ðuþ ¼ 1 JðuÞ, u 2M ð2:20þ (see Lemmas and of [8]). Denote by F the classes of symmetrc subsets of M and by (M) the Fadell Rabnowtz cohomologcal ndex of M 2F. Then þ k k :¼ nf sup M2F þ u2m ðmþk :¼ sup M2F ðmþk nf u2m þ ðuþ; 1 k ðm þ Þ; ðuþ; 1 k ðm Þ ð2:21þ defne non-decreasng (resp. non-ncreasng) sequences of postve (resp. negatve) egenvalues of (2.17) that are unbounded when (M ) ¼1 (see Theorems and of [8]). When (M ) ¼ 0 we set 1 ¼1for convenence. Returnng to (2.6), suppose that u ¼ 0 s a soluton and the asymptotc behavour of F near zero s gven by Fðu Þ¼ Jðu ÞþoðÞ as & 0, unformly n u 2M: ð2:22þ PROPOSITION 2.3 (Proposton of [8]) Assume (A 1 ) (A 3 ), (A 4 ) and J 2 C 1 (W, R) satsfy (2.18), J 0 and F 0 are compact, (2.22) holds, and zero s an solated crtcal pont of. () If þ 1, then Cq (,0) q0 2. () If kþ1 5 5 k or þ k 5 5 þ kþ1, then Ck (,0)6¼ Proof of Theorem 1.1 Frst we verfy that our problem fts nto the operator settng of Secton 2. Let W ¼ W 1,p (), A p u, v ¼ jru j p 2 ru rv þ a ðxþju j p 2 u v, ð3:1þ

6 578 S. El Manoun and K. Perera and FðuÞ ¼ Fðx, uþ: Then (A 1 ) s clear, ha p u, v ¼ ku k p n (A 2 ), and (A 3 ) holds wth I p ðu Þ¼ 1 jru j p þ a ðxþju j p : p By the Ho lder nequaltes for ntegrals and sums, A p u, v 1=p 0 jru j p 1=p 0 jru j p þ a ðxþju j p ¼ 1=p 1=p 0 jrv j p þ a ðxþju j p jrv j p þ a ðxþjv j p 1=p a ðxþjv j p 1=p ð3:2þ ð3:3þ ku k p 1 kv k, ð3:4þ so (A 4 ) follows from Proposton 2.1. By the growth condton (1.2),! F 0 X ðuþ, v m Xm X m ¼ F u ðx, uþv jjc ku j k r j 1 þ 1 kv L ðr j 1Þð p Þ0 k : j¼1 ð3:5þ Snce ðr j 1Þð p Þ0 5 p j and hence the mbeddng W 1, p j ðþ,! L ðr j 1Þð p Þ0 ðþ s compact, the compactness of F 0 follows. By (1.11), ðuþ Xm 1 ku k p jj C Xm p ku k t!: þ1 ð3:6þ Snce t 5p, t follows that s bounded from below and coercve for all. Then every PS sequence s bounded and hence satsfes the PS condton by Proposton 2.2. Turnng to the egenvalue problem (1.10), let JðuÞ ¼ Jðx, uþ, GðuÞ ¼ Gðx, uþ: ð3:7þ Then By (1.9), Jðu Þ¼ VðxÞjj r 1=p 1 þþr m =p m ju 1 j r 1 ju m j r m ¼jj JðuÞ: ð3:8þ jgðu Þj C Xm jj s =p k k s so (2.22) also holds. Applyng Proposton 2.3, we have C k (,0)6¼ 0. Proposton 1.2 now gves the result. u, ð3:9þ g

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