Existence and Multiplicity of Solutions for a Class of Semilinear Elliptic Equations 1

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1 Journal of Mathematical Analysis and Applications 257, (2001) doi: /jmaa , available online at on Existence and Multiplicity of Solutions for a Class of Semilinear Elliptic Equations 1 Shui-Qiang Liu Department of Mathematics, Shaoyang Teachers College, Shaoyang , People s Republic of China and Chun-Lei Tang Department of Mathematics, Southwest Normal University, Chongqing , People s Republic of China Submitted by J. McKenna Received March 6, 2000 The existence and multiplicity results are obtained for solutions of a class of the Dirichlet problem for semilinear elliptic equations by the least action principle and the minimax methods, respectively Academic Press Key Words: Dirichlet problem; resonance; critical point; critical growth; the least action principle; the minimax methods. 1. INTRODUCTION AND MAIN RESULTS Consider the Dirichlet boundary value problem u = f x u+hx for a.e. x u = 0on (1) where R N N 1 is a bounded domain, f R R is a Carathéodory function, that is, f x t is measurable in x for every t R 1 Project supported by the National Natural Science Foundation of China and Project 00C024 supported by the Educative Department of Hunan Province X/01 $35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved.

2 322 liu and tang and continuous in t for a.e. x, and h L 2. Letλ k k = 1 2 be the kth distinct eigenvalue of the eigenvalue problem u = λu in u = 0on and Eλ k k = 1 2 be the eigenspace corresponding to λ k. Set Fx t t 0 f x s ds for all t R and a.e. x. In this paper, we obtain an existence theorem by the least action principle for problem (1) in the critical growth case and a multiplicity result is obtained by using the minimax methods in the critical point theory, and in particular, a three-critical-point theorem proposed by Brezis and Nirenberg [4] in the subcritical growth case. The main results are the following theorems. Theorem 1. Suppose that there exist a constant C 1 > 0 and a real function γ L 1 such that f x t C 1 t γx (2) for all t R and a.e. x, where 2 2N/N 2 if N 3 and 2 may be replaced by any number in 1 + if N = 1 or 2, and that Fx t 1 2 λ 1t 2 (3) as t uniformly for a.e. x. Assume that h L 2 satisfies that hxψx dx = 0 (4) where ψ is the normalized eigenfunction corresponding to λ 1, ψx > 0 for all x. Then problem (1) has at least one solution in H0 1, where H1 0 is a Hilbert space with the norm given by ( ) 1 u = u 2 2 dx Theorem 2. Suppose that h = 0 (5) and that there exist C 2 > 0 and 2 <p<2n/n 2 for N 3(2 <p< +, for N = 1 2) such that f x t C 2 t p (6)

3 solutions for elliptic equations 323 for all t R and a.e. x. Assume that (3) holds and that there exists an integer m 1b>0, and δ>0 such that λ m f x t t λ m+1 b (7) for all 0 < t δ and a.e. x. Then problem (1) has at least two nonzero solutions in H 1 0. Many papers deal with the case that f x t lim = λ t t 1 (8) uniformly for a.e. x (see [1, 3, 5, 8, 10, 11, 13, 17 20] and their references), which is called resonant at the first eigenvalue. Note that (8) implies that for a.e. x, where A x =λ 1 A x lim sup t for a.e. x. The more general case that A x λ 1 2Fx t t 2 for a.e. x is considered by many authors (see [6, 7, 9, 14] and the references therein). The existence results are given for problem (1) in [1, 3, 5 11, 13, 14, 17 20]. In [2, 12, 21] some multiplicity theorems are obtained by using the topological degree technique and the variational methods, respectively. Except for [7, 14], the linear case is only treated. Reference [14] allows subcritical growth on f x t and in [7] the critical growth on f x t is considered. Remark 1 There are functions f x t and hx satisfying our Theorem 1 and not satisfying those in [1 3, 5 14, 16 20]. In fact, let f x t =λ 1 t 2t 1 + t t 2 1 cos t 2 and h L 2 satisfying (4). Then f x t does not satisfy the theorems in [1 3, 5, 6, 8 14, 16 20] for it is growing critically. Moreover f x t does not satisfy the theorems in [7] yet because A x =λ 1 in this case, { } inf v H 1 0 v L 2 =1 v 2 dx v 0 A xv dx = 0

4 324 liu and tang (see Theorem 1 in [7]). But Fx t = 1 2 λ 1t 2 ln1 + t 2 +sin t 2 satisfies (2) and (3). Hence f x t and hx satisfy our Theorem 1. Remark 2 There are functions f x t satisfying our Theorem 2 and not satisfying those in [2, 12, 21]. In fact, let { λm t t δ f x t = C 3 λ 1 t 2t/1 + t 2 +pt p 2 t cos t p t δ, where C 3 = λ m λ 1 2/1 + δ 2 +pδ p 2 cos δ p 1 and 2 <p<2n/ N 2 for N 3(2 <p<+, for N = 1 2). Then f x t satisfies our Theorem 2. But this f x t does not satisfy the theorems in [2, 12, 21] for it is not linearly growing. Let ϕu = PROOF OF THEOREMS u 2 dx Fx u dx hu dx for u H0 1. In a way similar to Theorem 1.4 in [15] we can prove that ϕ C 1 H0 1R. It is well known that u H1 0 is a solution of problem (1) if and only if u is a critical point of ϕ. Foru H0 1, set ( ) ū = u ψdx ψ ũ = u ū Then one has ū Eλ 1 ũ Eλ 1 Proof of Theorem 1 First there exist a real function g L 1, and G CR R which is subadditive, that is, Gs + t Gs+Gt (9) for all s t R, and coercive, that is, Gt + (10) as t, and satisfies that Gt t+4 (11)

5 for all t R, such that solutions for elliptic equations 325 Fx t 1 2 λ 1t 2 Gt+gx (12) for all t R and a.e. x. In fact, since Fx t 1 2 λ 1t 2 as t uniformly for a.e. x, there exists a sequence of positive integers (n k ) with n k+1 > 2n k for all positive integer k such that Fx t 1 2 λ 1t 2 k (13) for all t n k and a.e. x. Letn 0 = 0 and define Gt =k t n k 1 n k n k 1 (14) for n k 1 t <n k, where k N. By the definition of G we have for n k 1 t <n k. It follows that for all t R and a.e. x, where k + 2 Gt k + 3 (15) Fx t 1 2 λ 1t 2 Gt+gx gx = C 1 2 n2 1 + n 1γx+ λ 1 2 n2 1 + n 1+4 In fact, by (2) one has Fx t 1 2 λ 1t 2 C 1 2 t2 + γxt+ λ 1 +t (16) 2 t2 for all t R and a.e. x. For every t R there exists k N such that In the case that k = 1 we have n k 1 t <n k Fx t 1 2 λ 1t 2 C 1 2 n2 1 + γxn 1 + λ 1 2 n2 1 + n 1 = gx 4 Gt+gx for all t n 1 and a.e. x by (16) and (15). In the case that k 2, one has Fx t 1 2 λ 1t 2 k 1 k + 3+gx Gt+gx for all n k 1 t n k and a.e. x by (13) and (15).

6 326 liu and tang It is obvious that G is continuous and coercive. Moreover one has Gt t+4 for all t R. In fact, for every t R there exists k N such that which implies that n k 1 t <n k Gt k + 3 n k t+4 for all t R by (15) and the fact that n k k for all integers k 0. Now we only need to prove the subadditivity of G. Let and m = maxk j. Then we have Hence we obtain, by (15), n k 1 s <n k n j 1 t <n j s + t s+t <n k + n j 2n m <n m+1 Gs + t m + 4 k j + 2 Gs+Gt which shows that G is subadditive. Second, the functional Gv dx is coercive on Eλ 1. If not, there exist C 0 > 0 and a sequence v n in Eλ 1 such that v n and Gv n dx C 0, which implies that lim sup n Gv n dx C 0 < + (17) It follows from the first part of the proof of Lemma 3.2 in [3] that for every α>0 there exists m α > 0 such that measx vx <m α v <α for all v Eλ 1. For the convenience of the reader we state another proof. If it does not hold, there exist α 0 > 0 and a sequence v n n=1 Eλ 1 such that { meas x v n x < 1 } n v n α 0 for all n, which implies that v n 0 for all n. By the homogeneity of the above inequality we may assume that v n =1 and { meas x v n x < 1 } α n 0

7 solutions for elliptic equations 327 for all n. It follows from the compactness of the unit sphere of Eλ 1 that there exists a subsequence, say v n, such that v n converges to some v in Eλ 1. Hence v 0 and v n v 0asn by the equivalence of the norms on the finite-dimensional space Eλ 1. For every positive integer m, there exists N 2m such that v n v < 1/2m whenever n N. Hence one has { x v N x < 1 } { x vx < 1 } N m which follows from the inequality vx v N v +v N x 1 2m + 1 N 1 m for all x with v N x < 1/N. Thus we have { meas x vx < 1 } α m 0 for all m, which implies that v = 0 on a positive measure subset. It contradicts the unique continuation property of the eigenfunction. Let A n =x v n x m α v n Then one has meas\a n <α. For every β>0, there exists M>0 such that Gt β for all t M by the coercivity of G. Let For x A n, one has B n =x v n x M v n x m α v n which implies that A n B n for large n. Now we have, by (11), Gv n dx β meas B n M + 4 meas \B n β meas A n M + 4 meas \A n βmeas α M + 4α for large n. Hence one has lim inf Gv n dx βmeas α M + 4α n

8 328 liu and tang Letting α 0, we obtain lim inf n By the arbitarity of β we have lim inf n Gv n dx β meas Gv n dx =+ which contradicts (17). Third, the functional ϕ is coercive. In fact, from (12), (9), and (11) one obtains Fx u dx 1 2 λ 1 u 2 dx Gu dx + gx dx Gū G ũ dx + gx dx Gū dx +ũ L 1 + 4meas + Gū dx + C 4 ũ+1 for all u H0 1 and some C 4 = C + 4 meas + gx dx where C is a positive constant in Sobolev s inequality, gx dx u L 1 Cu u L 2 Cu for all u H0 1. Hence we have ϕu= 1 u 2 dx 1 ( 2 2 λ 1 u 2 dx 1 ũ 2 dx λ 1 1 ( 1 λ ) 1 2 λ 2 Fxudx 1 2 λ 1 ũ 2 dx+ Gūdx C 4 ũ+1 ũ 2 dx C 5 ũ+1+ Gūdx ) u 2 dx hudx hudx for all u H 1 0 and some C 5 = C 4 + Ch L 2, which implies that ϕ is coercive by the coercivity of the functional Gv dx on Eλ 1 and the fact that u 2 =ū 2 +ũ 2

9 solutions for elliptic equations 329 At last, the functional ϕ is weakly lower semicontinuous. Indeed, from (12) and (15) we obtain Fx t 1 2 λ 1t 2 + gx for all t R and a.e. x. In a way similar to the first part of the proof of Theorem 1 in [7], one can prove that ϕ is weakly lower semicontinuous. It follows from the least action principle (see, e.g., Theorem 1.1 in [15]) that ϕ has a minimum. Hence problem (1) has at least one solution in H 1 0, which completes our proof. Now we prove Theorem 2. For the convenience of the reader we state a three-critical-point theorem proposed by Brezis and Nirenberg (see Theorem 4 in [4]). Lemma 1 [4]. Let X be a Banach space with a direct sum decomposition X = X 1 X2 with dim X 2 < and let ϕ be a C 1 function on X with ϕ0 =0, satisfying the (PS) condition. Assume that, for some δ 0 > 0, and ϕv 0 for v X 1 with v δ 0 ϕv 0 for v X 2 with v δ 0 Assume also that ϕ is bounded from below and inf X ϕ<0. Then ϕ has at least two nonzero critical points. Proof of Theorem 2. Let X 2 be a finite-dimensional subspace of X = H0 1 given by X 2 = Eλ 1 Eλ m and let X 1 = X2. Then there exists δ 0 > 0 such that ϕu 0 ϕu 0 In fact, from (7) one obtains for all u X 2 with u δ 0 for all u X 1 with u δ 0 λ m t 2 tf x t λ m+1 bt 2 for all t <δand a.e. x, which implies that λ m t 2 s tf x st λ m+1 bt 2 s (18) for all 0 < s 1 t < δ and a.e. x. Noting that Fx t = 1 0 tf x stds, we obtain 1 2 λ mt 2 Fx t 1 2 λ m+1 bt 2

10 330 liu and tang for all t <δand a.e. x. By the equivalence of the norms on the finite-dimensional space X 2 there exists C 6 > 0 such that u C 6 u for all u X 2. Hence one has ϕu 1 u 2 dx λ m for all u X 2 with u δ/c 6 by (5). By (6) we have Fx t C 2 p 1 t p +t for all t R and a.e. x. Thus one has u 2 dx 0 Fx t C 2 p 1 + δ 1 p t p = C7 t p for all t δ and a.e. x. Hence we obtain, by (7), Fx t 1 2 λ m+1 bt 2 + C 7 t p for all t R and a.e. x. It follows from (5) that ϕu 1 u 2 dx λ m+1 b u 2 dx C 7 u p L p b u 2 C 2λ 7 C p u p m+1 0 for all u X 1 with u b/2λ m+1 C 7 C p 1/p 2, where C>0 is a constant such that u L p Cu for all u H0 1 by Sobolev s embedding theorem. It is obvious that ϕ is a C 1 -function with ϕ0 =0. By the proof of Theorem 1, ϕ is coercive and bounded from below. Thus ϕ satisfies the (PS) condition by the subcritical growth of f x t and the coercivity of ϕ. In the case that inf X ϕ<0, Theorem 2 follows from Lemma 1. In the case that inf X ϕ 0, by (18) one has ϕu =inf X ϕ = 0 for all u X 2 with u δ, which implies that all u X 2 with u δ are solutions of problem (1). Therefore Theorem 2 is proved. ACKNOWLEDGMENT The authors thank the referee for valuable suggestions.

11 solutions for elliptic equations 331 REFERENCES 1. S. Ahmad, A resonance problem in which the nonlinearity may grow linearly, Proc. Amer. Math. Soc. 92 (1984), S. Ahmad, Mutiple nontrival solutions of resonant and nonresonant asymptotically linear problems, Proc. Amer. Math. Soc. 96 (1986), P. Bartolo, V. Benci, and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal. 7 (1983), H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), P. Drábek, On the resonance problem with arbitrary linear growth, J. Math. Anal. Appl. 127 (1987), D. G. De Figueiredo and J.-P. Gossez, Nonresonance below the first eigenvalue for a semilinear elliptic problem, Math. Ann. 218 (1988), J. V. A. Goncalves, On nonresonant sublinear elliptic problems, Nonlinear Anal. 15 (1990), C. P. Gupta, Solvability of a boundary value problem with nonlinearity satisfying a sign condition, J. Math. Anal. Appl. 129 (1988), A. Hammerstein, Nichtlineare integralgleichungen nebst anwendungen, Acta Math. 54 (1930), R. Iannacci and M. N. Nkashama, Nonlinear two point boundary value problem at resonance without Landesman Lazer condition, Proc. Amer. Math. Soc. 10 (1989), E. Landesman and A. Lazer, Nonlinear perturbation of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), E. Landesman, S. Robinson, and A. Rumbos, Multiple solutions of semilinear elliptic problems at resonance, Nonlinear Anal. 24 (1995), J. Mawhin, J. R. Ward, and M. Willem, Necessary and sufficient conditions for the solvability of a nonlinear two-point boundary value problem, Proc. Amer. Math. Soc. 93 (1985), J. Mawhin, J. R. Ward, and M. Willem, Variational methods and semilinear elliptic equations, Arch. Rational Mech. Anal. 95 (1986), J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer- Verlag, New York, P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in CBMS Regional Conf. Ser. in Math., Vol. 65, Amer. Math. Soc., Providence, C. L. Tang, Solvability for two-point boundary value problems, J. Math. Anal. Appl. 216 (1997), J. R. Ward, A boundary value problem with a periodic nonlinearity, Nonlinear Anal. 10 (1986), S. Solimini, On the solvability of some elliptic partial differential equation with the linear part at resonance, J. Math. Anal. Appl. 117 (1986), D. Lupo and S. Solimini, A note on a resonance problem, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), A. Capozzi, D. Lupo, and S. Solimini, On the existence of a nontrivial solution to nonlinear problems at resonance, Nonlinear Anal. 13 (1989),