Remarks on Multiple Nontrivial Solutions for Quasi-Linear Resonant Problems

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1 Journal of Mathematical Analysis and Applications 258, ) doi:0.006/jmaa , available online at on Remarks on Multiple Nontrivial Solutions for Quasi-Linear Resonant Problems Jiaquan Liu Academy of Mathematics Science, Peking University, Beijing 0087, People s Republic of China and Jiabao Su 2 Department of Mathematics, Capital Normal University, Beijing 00037, and Academy of Mathematics and Systems Science, Beijing 00080, People s Republic of China Submitted by Catherine Bandle Received July 7, 2000 In this paper Morse theory and local linking are used to study the existence of multiple nontrivial solutions for a class of Dirichlet boundary value problems with double resonance at infinity and at Academic Press Key Words: Quasi-linear elliptic equation; double resonance; critical group; homological nontrivial critical point; Morse theory; local linking; multiple solutions.. INTRODUCTION In this paper we consider the existence of multiple nontrivial solutions of the Dirichlet boundary value problem { p u = f x u in D u = 0 on, p where N is a bounded open domain with smooth boundary and f ω is a Carathéodory function with the subcritical growth f x t c +t q a.e. x t.) Supported by the National Natural Science Foundation of China. 2 Supported in part by the National Natural Science Foundation of China, the Natural Science Foundation of Beijing, and the Foundation of Beijing s Educational Committee X/0 $35.00 Copyright 200 by Academic Press All rights of reproduction in any form reserved.

2 20 liu and su where q Np/N p if <p<n and q + if <N p. Here p denotes the p-laplacian operator; that is, p u = div u p 2 u. When p = 2, it is the usual Laplacian operator. From a variational stand point, finding weak solutions of D p in W p 0 is equivalent to finding critical points of the C functional given by Ju = p u p dx Fx u dx u W p 0.2) where Fx t = t p 0 f x sds and the Sobolev space W0 is a Banach space endowed with the norm u = up dx p. Let f x 0 0; then D p admits the trivial solution u 0. We are interested in finding nontrivial solutions for D p. The existence of nontrivial solutions for D p depends on the behaviors of the term f x t or its primitive Fx t near 0 and near infinity. It is well known See [8]) that the p-homogeneous boundary value problem { p u = λu p 2 u in D 0 p u = 0 on has the first eigenvalue λ > 0 that is simple and has an associated eigenfunction which is positive in. It is also known that λ is an isolated point of σ p, the spectrum of p, which contains at least an increasing eigenvalue sequence obtained by the Lusternik Schnirlaman theory. Let V =ϕ be the one-dimensional eigenspace associated to λ, where ϕ > 0in and ϕ =. Taking one subspace W W p 0 complementing V such that W p 0 =V W, there exists λ >λ such that u p dx λ u p dx for u W.3) When p = 2, one can take λ = λ 2, the second eigenvalue of in H 0. In this paper we give conditions on which D p has at least two nontrivial solutions. More precisely, we have the following results. Theorem.. Let f satisfy the following conditions: f 0 ) There is some r>0 small and λ < ˆλ < λ such that λ t p pfx t ˆλt p for t with t r a.e. x f ) lim sup t pfxt t p <λ Then the problem D p has at least two nontrivial solutions in W p 0.

3 Theorem.2. multiple nontrivial solutions 2 Let f satisfy f 0 ) and let the following conditions hold: f 2 ) lim t pfxt t p = λ. f 3 ) lim t f x tt pfx t=+. Then the problem D p has at least two nontrivial solutions in W p 0. Remark.. i) In Theorems. and.2, the condition f 0 ) allows the problem D p ) is resonant near 0 at the first eigenvalue λ from the right side. Clearly, f 0 ) contains the situations lim t 0 pfx t/t p = λ λ λ. But here we do not need to assume that the limit exists. ii) f is a nonresonance condition, and f 2 means the problem D p is resonant at infinity. In Theorem.2, the more interesting case is that near the 0; near infinity, the problem D p is resonant at the same eigenvalue λ. iii) The same conclusions hold if we replace the conditions f 0, f, and f 2 by the following stronger conditions, respectively: f 0 ) There is some r>0 small and λ < ˆλ < λ such that λ t p f x tt ˆλt p for t with t r a.e. x f xt f ) lim sup t <λ t p 2 t. f 2 ) lim f xt t = λ t p 2 t. The problem D p at resonance with p 2 has been studied by a few authors. With various conditions imposed on f x t or Fx t and via directly variational methods or the minimax method, such as the well-known saddlepoint theorem or the mountain pass theorem see [2]), solvability results for one solution were obtained see, e.g., [ 3, and ]). All of them treated the situation for resonance at infinity. Thélin [23] discussed D p ) or similar problems and obtained some results of the existence and nonexistence of positive solutions. More recently, Li and Zhou [7] studied the existence of positive solution for the case where f x t =0 for t 0 and lim t + f x t/t = l>λ via a variant of the mountain pass theorem. To our knowledge, no multiplicity results have been obtained in the literature for the resonance case or for the nonresonance case. So our results are new and, moreover, as we show our methods are different from the aforementioned methods. Theorem. complements one of main results of Coast and Magalhaes [], who obtained the existence of at least one nontrivial solution for the case pfx t pfx t lim sup α<λ t 0 t p <β lim inf a.e. x t t p These results are new even for the linear case p = 2, because we require only that f x t be a Carathédory function with subcritical growth.

4 22 liu and su Under these conditions, the corresponding functional defined by.2) is only of C, and no Morse indices are concerned. Moreover, based on the nice properties of σ, the spectrum of, we can unify the conditions f and f 2.LetEλ k denote the eigenspace associated to λ k. Then we have the following stronger results. Theorem.3. Let f satisfy the following: f 4 ) There is some r>0 small such that λ k t 2 2Fx t λ k+ t 2 t t r a.e. x k f 5 ) lim sup t f xt t λ. f 6 ) Write gx t =f x t λ t.ifu n and v n /u n, then there exist δ>0 and N 0 such that gx u nv n dx δ, where u n = v n + w n and v n Eλ, w n Eλ. Then the problem D 2 has at least two nontrivial solutions in H 0. Remark.2. The condition f 4 allows the problem D 2 is resonant near 0 between any two consecutive eigenvalues of in H0. It is clear that the conclusion is valid if we replace f 4 by the following condition: f 4 ) There is some r>0 small such that λ k t 2 f x tt λ k+ t 2 t t r a.e. x k Theorem.3 extends some results in [2 and 22], where it was required that f x 0 λ k λ k+, k 0λ 0 =, and the results obtained via the Leray Schauder degree which cannot be applied in our case. Theorem.3 also extends some results of Chang [8, 9] and others. The conditions presented here are very intuitive, and the conclusions are interesting. The existence of multiple solutions depends mainly on the local behavior of f x t or Fx t near 0 and near infinity. Without these conditions, it is hard to get multiple solutions of D p even for the case where p = 2. The proofs of our theorems will concern the Morse theory and critical groups [8, 20]. Few applications of this method has been carried in treating the quasi-linear problems. In next section we present some abstract results. 2. PRELIMINARY: THREE CRITICAL POINT THEOREM The Morse theory is known to be very useful in studying the existence of multiple solutions of differential equations having the variational structure. The concept of critical groups is particularly pertinent.

5 multiple nontrivial solutions 23 Let X be a real Banach space and let J C X, =u X J u =0. Letu be an isolated critical point of J with Ju =c, and U be a neighborhood of u, containing the unique critical point, the group C q J u=h q J c U J c U\u q = ) is called the q-th critical group of J at u, where J c =u X Ju c and H q the q-th singular relative homology group with integer coefficients. We say that u is an homological nontrivial critical point of J if at least one of its critical groups is nontrivial. J satisfies the deformation condition D c [5, 6]) at level c if for any ε >0 and any neighborhood of c, there are ε>0 and a continuous deformation η X 0 X such that i) ii) ηu t =u for either t = 0oru J c ε c + ε Jηu t is nonincreasing in t for any u X iii) ηj c+ε \ J c ε. We say that J satisfies D if J satisfies D c for all c. We note here that the usual PS condition or the C condition introduced by Cerami see [4]) can imply the D condition. Let J satisfy the D condition and let # <. Take a<inf J. By the Morse theory, the information of all critical point of J are contained in the Morse inequality Pt u =Pt + + tqt 2.2) u where Ptu= dimc q Ju t q and Pt = dimh q XJ a t q 2.3) q=0 are the Poincaré polynomials of J at u and at infinity and Qt is a formal series with nonnegative integer coefficients. Now we present the following abstract critical point theorem. Theorem 2.. Let X be a real Banach space and let J C X satisfy the D condition and be bounded from below. If J has a critical point that is homological nontrivial and is not the minimizer of J, then J has at least three critical points. Proof. The idea of its proof is similar to that of in [9]. We sketch it for the interested reader s convenience. Because J is bounded from below and satisfies the deformation property, it follows that J attains its minimum at some u X and C q J u = δ q0. Take a<inf JX; then q=0

6 24 liu and su H q X J a = δ q0. Of course, we may assume that u is the only minimizer of J. Assume that =u u, where u is the known critical point of J. Then the Morse inequality 2.2) reads as Pt u +Pt u =Pt + + tqt 2.4) It follows that Pt u = + tqt. Because u is not a minimizer of J and is homologically nontrivial, we see that C 0 J u = 0 and Pt u 0. Hence there exists q 2 such that H q J c J c \u = C q J u = 0 where c = Ju. By the deformations X J c and J c \u u,we have H q J c = H q X = 0 H q J c \u = H q J c \u = 0 Now from the exact sequence H q J c \u Hq J c \u i Hq J c j Hq J c J c \u we see easily that H q J c J c \u = C q J u = 0. This contradiction completes the proof. Theorem 2. is a slightly modification of Theorem 2.2 in Liu [9]. We mention that the condition that J has an homological nontrivial critical point differing from the global minimizer is essential in this theorem. In applications one often has the trivial critical point u = 0 and describes the critical groups C q J 0 in order to finding nontrivial critical point. It was showed in [9] that under some local conditions near 0, 0 seems to be homological nontrivial. Let us recall this result below. Proposition 2. [9]. Assume that J has a critical point u = 0 with J0 =0. IfJ has a local linking at 0 with respect to X = V W, k = dim V<, i.e., there exists ρ>0 small such that Ju 0 u V u ρ Ju > 0 u W 0 < u ρ 2.5) Then C k J 0 = 0; that is, 0 is an homological nontrivial critical point of J. The concept of local linking was introduced by Li and Liu [3] and plays important role in finding nontrivial critical points. It is a weaker condition that makes the trivial critical point is homologically nontrivial. Most of its applications have been carried in treating semilinear problems see e.g., [4 6]). Later we apply it to the quasi-linear case and use it to prove our existence result.

7 multiple nontrivial solutions PROOF OF THEOREMS. AND.2 In this section we prove Theorems. and.2 via the abstract results in Section 2. Lemma 3.. Let f satisfy.). Then any bounded sequence u n W p 0 such that J u n 0 in W p 0 as n has a convergent subsequence. Proof. The proof is standard. Let u n W p 0 be bounded and such that J u n 0inW p 0 as n. Up to a subsequence, we may assume that there is some u 0 W p 0 such that u n u 0 in W p 0 and u n u 0 in L p 3.) It follows from J u n 0 that p u n f x u n 0 in W p 0 as n 3.2) Because p is an homeomorphism from W p 0 to W p 0,wesee that u n p f x u n 0 in W p 0 as n 3.3) By.), the map u f x u is completely continuous from W p 0 to W p 0, and it follows that p f x u n p f x u 0 in W p 0 3.4) Therefore, u n u 0 in W p 0. Lemma 3.2. If f satisfies either f )orf 2 ) and f 3, then i) ii) J is coercive on W p 0 ; that is, Ju + as u J satisfies the PS condition. Proof. i. a) Let f ) hold. It follows from f ) and.) that for some ε>0 small, there is constant C>0 such that Fx t p λ εt p + C t a.e. x 3.5) Therefore by the Poincaré inequality, for u W p 0, Ju= u p dx Fxudx p p up p λ εu p L C p λ ) ε u p C + as u 3.6) p λ

8 26 liu and su b) Let f 2 ) and f 3 ) hold. Write Fx t = p λ t p + Gx t and f x t =λ t p 2 t + gx t. Then pgx t lim = 0 and lim gx tt pgx t=+ 3.7) t t p t It follows that for every M>0, there is R M > 0 such that gx tt pgx t M t t R M a.e. x 3.8) Integrating the equality [ ] d Gx t = dt t p over the interval t T R M +, we have Gx T T p Gx t t p gx tt pgx t t p+ 3.9) M p T ) 3.0) p t p Letting T +, we see that Gx t M/p, for t, t R M, a.e. x. In a similar way, we have Gx t M/p for t, t R M, a.e. x. Hence lim Gx t = a.e. x 3.) t Let u n W p 0 be such that u n as n and Ju n Ĉ for some constant Ĉ. Taking v n = u n /u n, then up to subsequence, we may assume that there is some v 0 W p 0 such that Now So v n v 0 Ĉ u n p Ju n u n p = p in W p 0 v n v 0 in L p and v n x v 0 x a.e. on p 3.2) v n p λ v n p dx Gx u u n p n dx v n p λ v n p dx + M pu n p Gx u u n p n dx u n x R M v p n p λ v n p dx C 3.3) u n p lim sup v n p dx λ v 0 p dx 3.4) n

9 multiple nontrivial solutions 27 Because the norm is weakly semicontinuous, using the Poincaré inequality again, we have λ v 0 p dx v 0 p dx lim inf v n p dx n lim sup v n p dx 3.5) n By 3.4) and 3.5), v 0 p dx = λ v 0 p dx and v n v 0 in W p 0 with v 0 =. Hence v 0 =±ϕ. Take v 0 = ϕ ; then u n x + a.e. on. SoGx u n x a.e. on by 3.). Therefore, Ĉ Gx u n x dx + as n 3.6) This is impossible; hence J is coercive on W p 0. ii) The PS) condition follows from i) and Lemma 3.. Lemma 3.3. Let f satisfy f 0 then J has the local linking 2.5) at the origin with respect to W p 0 =V W, k = dim V =. Proof. i) Take u V. Since V is finite dimensional, it is easily seen that u ρ ux r x for ρ>0 small. So it follows from f 0 that for u ρ, Ju = p = λ p = u r u p dx Fx u dx u p dx [ λ p up Fx u Fx u dx ] dx 0 3.8) ii) Take u W, using.) and.3) we have the following estimates. Ju= u p dx Fxudx p = [ u p ˆλu p dx Fxu ˆλ ] p u r p up dx [ Fxu ˆλ ] u>r p up dx ˆλ p λ up c u s dx ˆλ p λ up cu s p<s p =Np/N p 3.9)

10 28 liu and su It follows that when u W and 0 < u ρ for ρ>0 small, Ju > 0. This completes the proof. Proof of Theorems and 2. By Lemma 3.2, J is coercive and satisfies the PS) condition and then the D condition. Hence J is bounded from below. By Lemma 3.3 and Proposition 2., the trivial solution u = 0is homological nontrivial and is not a minimizer. The conclusion follows from Theorem PROOF OF THEOREM.3 Theorem.3 follows from the following lemmas, Proposition 2., and Theorem 2.. Lemma 4.. If f satisfies f 5 ) and f 6 ), then the functional J is coercive on H0 and then satisfies the PS) condition. Proof. Write J as Ju = 2 u 2 dx 2 λ u 2 dx Gx u dx 4.) Write u = v + w with v Eλ and w Eλ.Byf 5 ), we see that for given ε>0, there is ξ>0 such that gx tt εt 2 for t ξ. Hence for s 0, J suu=s u 2 λ u 2 dx gxsuudx s λ ) w 2 gxsuudx gxsuudx λ 2 su>ξ su ξ s λ ) w 2 s εu 2 dx c λ 2 su ξ s λ ) w 2 εs u 2 c 4.2) λ 2 λ So Ju = = 2 J suuds [ s λ λ λ 2 λ 2 ) w 2 ε ] u 2 c ds λ ) w 2 λ λ 2 ε λ ε 2λ u 2 c ) w 2 ε 2λ v 2 c 4.3)

11 multiple nontrivial solutions 29 Take 0 <ε<λ λ 2 λ /λ 2. It is clear that for fixed R>0, Ju + as u with v =R 4.4) Because J is weakly lower semicontinuous on H0, it follows that J attains its minimum on the set C R =u H0 u = v + w v =R 4.5) That is, for any given R>0, there is u R H0 such that Ju R Ju for u C R 4.6) Now take u n H 0 such that u n and Ju n Ju for u C τn 4.7) where we write u n = w n + τ n ϕ for the sake of convenience and ϕ > 0is the first eigenfunction such that ϕ =. If there is some θ 0 such that τ n θu n for large n, then w n θ 2 τ n, and so Ju n 2 λ λ 2 ε λ ) w n 2 ε 2λ θ 2 w n 2 c 4.8) and it follows that Ju n + as n if ε>0 is small enough. Therefore, we consider the case where τ n /u n asn.uptoa subsequence, we may choose τ n < τ n+. Now where Ju n+ Ju n =Jw n+ + τ n+ ϕ Jw n + τ n ϕ Jw n+ + τ n+ ϕ J w n+ + τ ) n τ n+ τ n+ϕ [ = G x w n+ + τ ) n τ n+ τ n+ϕ ] Gx w n+ + τ n+ ϕ dx = 0 gx w n+ + τ n tτ n+ ϕ ) τ τ n tτ n+ ϕ dx n t dt 4.9) τ n t τ n t = tτ n+ tτ n+ 4.0) τ n+

12 220 liu and su Because τ n tτ n+ ϕ as n 4.) w n+ + τ n tτ n+ ϕ it follows from f 6 that there is δ>0 and N 0 such that gx w n+ + τ n tτ n+ ϕ τ n tτ n+ ϕ dx δ for n N 0 4.2) Hence τ n Ju n+ Ju n δ t 0 = δ 0 τ n t dt τ n τ n+ tτ n + t τ n+ dt = δ lntτ n + t τ n+ 0 = δln τ n+ ln τ n 4.3) So n Ju n+ =Ju N0 + Ju i+ Ju i i=n 0 n Ju N0 +δ ln τ i+ ln τ i i=n 0 = Ju N0 +δln τ n+ ln τ N0 as n 4.4) Therefore, J is coercive on H 0. Lemma 4.2. Let f satisfy f 4 ). Then functional J has a local linking 2.5) with respect to H 0 =V W, where V = k j= Eλ j and W = j k+ Eλ j. Proof. i) Because V is finite dimensional, we have that for given r>0, there is some ρ>0 such that u V u ρ ux r/3 r a.e. x 4.5) Now on V, we have by f 4 that for u V with u ρ, [ ] Ju = 2 u2 Fx u dx u 2 λ 2 k u 2 dx 0 4.6)

13 multiple nontrivial solutions 22 ii) For u W, we write u = v + w, where v Eλ k+ and w j>k+ Eλ j. Then Ju λ ) ) k+ w λ k+2 2 λ k+u 2 Fx u dx 4.7) For ux r, ux r ) 2 λ k+u 2 Fx u dx 0 4.8) For ux >r, we have wx 2 ux. Hence by.) and the Poincaré 3 inequality, we have Hence ux>r 2 λ k+u 2 Fxu Ju 2 + ) dx c λ ) k+ w 2 ux r λ k+2 ux>r ux s+ dx c 3/2 s+ wx s+ dx cw s+ where 2 <s ) ) 2 λ k+u 2 Fx u dx cw s+ 4.20) If w = 0v 0, and v ρ, then vx r when ρ is small enough. So ) Jv = 2 v2 Fx v dx ) = 2 λ k+ v 2 Fx v dx 0 4.2) If Jv =0, then by f 4 ), f x t =λ k+ t a.e. on and t r. Back to D 2, we find that 0 is not an isolated critical point of J. So we get the conclusion that This completes the proof. Ju > 0 for u W with 0 < u ρ 4.22) ACKNOWLEDGMENTS The authors thank Professor Liu Zhaoli for many valuable discussions and suggestion. They also thank the referee for his valuable comments and suggestions.

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