MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRÖDINGER-POISSON EQUATIONS WITH SIGN-CHANGING POTENTIAL. Lixia Wang. Shiwang Ma. Na Xu

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1 MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRÖDINGER-POISSON EQUATIONS WITH SIGN-CHANGING POTENTIAL Lixia Wang School of Sciences Tianjin Chengjian University, Tianjin 0084, China Shiwang Ma School of Mathematical Sciences and LPMC Nankai University, Tianjin 0007, China Na Xu School of Sciences Tianjin University of Technology and Education, Tianjin 00222, China Abstract. In this paper, we study the following nonhomogeneous Schrödinger- Poisson equations { u + λv (x)u + K(x)φu = f(x, u) + g(x), x, φ = K(x)u 2, x, where λ > 0 is a parameter. Under some suitable assumptions on V, K, f and g, the existence of multiple solutions is proved by using the Ekeland s variational principle and the Mountain Pass Theorem in critical point theory. In particular, the potential V is allowed to be sign-changing.. Introduction and main results In this paper we consider the following Schrödinger-Poisson equations { u + λv (x)u + K(x)φu = f(x, u) + g(x), x, (SP ) λ φ = K(x)u 2, x, where λ is a parameter, V C(, R) and f C( R, R). Such a system, also called Schrödinger-Maxwell equations, arises in an interesting physical context. In fact, according to a classical model, the interaction of a charged particle with an electromagnetic field can be described by coupling the nonlinear Schrödinger s and Poisson s equations (we refer reader to [6, 2] and the references therein for more details on the physical aspects). In particular, if we are looking for electrostatic-type solutions, we just have to solve system (SP ) λ. Variational methods and critical point theory are powerful tools in studying nonlinear differential equations [9, 25, ], and in particular Hamiltonian system [26, 27], and also impulsive Hamiltonian systems [2]. In recent years, (SP ) λ has been studied widely via modern variational methods under various hypotheses on 200 Mathematics Subject Classification. 5B, 5J65,5Q55. Key words and phrases. Nonhomogeneous; Schrödinger-Poisson equations; sign-changing potential; Ekeland s variational principle; Mountain Pass Theorem. Research supported by the Specialized Fund for the Doctoral Program of Higher Education and the National Natural Science Foundation of China.

2 2 LIXIA WANG, SHIWANG MA AND NA XU the potential V and the nonlinear term f, see [, 4, 6, 0,, 2,, 9] and the references therein. We recall some of them as follows. The case of g 0, that is the homogeneous case, has been studied widely in [, 8, 9, 2,, 8, 20, 2, 24] when V is a constant or radially symmetric, and in [, 40] when V is not radially symmetric. Very recently, Azzollni and Pomponio in [4] proved the existence of a ground state solution for the following system { u + V (x)u + φ(x)u = f(x, u), x, (.) φ = u 2, x, with f(x, u) = u p 2 u(2 < p < 6) and non-constant potential V which may be unbounded below; When V (x) and f(x, u) are -periodic in each x i, i =, 2,, Zhao et al. [9] obtained the existence of infinitely many geometrically distinct solutions by using the nonlinear superposition principle established in []. Zhao et al. [4] considered the existence of nontrivial solution and concentration results of (SP ) λ provided that V satisfies: (V 0 ) There is b > 0 such that meas{x : V (x) b} < +, where meas denotes the Lebesgue measures; (V ) V C(, R) and V is bounded below; (V 2 ) Ω = intv (0) is nonempty and has smooth boundary and Ω = V (0). This kind of hypotheses was first introduced by Bartsch and Wang [5] in the study of a nonlinear Schrödinger equation and the potential V (x) with V satisfying (V 0) (V 2) is referred as the steep well potential. Sun, Su and Zhao [29] got infinitely many solutions under suitable assumptions. Wu [5] studied the combined effect of concave and convex nonlinearities on the number of solutions for a semi-linear elliptic equation. For more results on the effect of concave and convex terms of elliptic equations see [6, 7] and the reference therein. In 204, Ye and Tang [8] studied the existence and multiplicity of solutions for homogeneous system of (SP ) λ when the potential V may change sign and the nonlinear term f is superlinear or sublinear in u as u. For the Schrödinger- Poisson system with sign-changing potential see [0] and sublinear Schrödinger- Poisson system see [28]. Next, we consider the nonhomogeneous case, that is g 0. The existence of radially symmetric solutions was obtained for above nonhomogeneous system with λ and K(x) in [24]. Chen and Tang [] obtained two solutions for the nonhomogeneous system with f(x, u) satisfying Amborosetti-Rabinowitz type condition and V being nonradially symmetric. In 205, Wang and Ma [2] considered the nonhomogeneous Schrödinger-Poisson equation containing concave and convex terms. For more results on the nonhomogeneous case see [4, 6, 7, 42] and the reference therein. Motivated by the above works, in the present paper we consider system (SP ) λ with more general potential V (x), K(x) and f(x, u). Under (V 0 ) (V ) and some more generic 4-superlinear conditions on f(x, u), we prove the existence of multiple solutions of problem (SP ) λ when λ > 0 large by using variation method. Precisely, we make the following assumptions. (f ) F (x, u) = u f(x, s)ds 0 for all (x, u) and f(x, u) = o(u) uniformly in x 0 as u 0.

3 SCHRÖDINGER-POISSON SYSTEMS (f 2 ) F (x, u)/u 4 + as u + uniformly in x. (f ) F(x, u) := 4 f(x, u)u F (x, u) 0 for all (x, u) R R. (f 4 ) There exist a, L > 0 and τ (/2, 2) such that f(x, u) τ a F(x, u) u τ, for all x and u L. (K) K(x) L 2 ( ) L ( ) and K(x) 0 is not identically zero for a.e. x. (g) g(x) L 2 ( ) and g(x) 0 for a.e. x. Remark It follows (f 2 ) and (f 4 ) that f(x, u) τ a 4 f(x, u) u τ+ for large u. Thus, by (f ), for any ε > 0, there exists C ε > 0 such that (.2) and (.) f(x, u) ε u + C ε u p, (x, u) R F (x, u) ε u 2 + C ε u p, (x, u) R, where p = 2τ/(τ ) (4, 2 ), 2 = 6 is the critical exponent for the Sobolev embedding in dimension. Before stating our main results, we give several notations. Let H ( ) be the usual Sobolev space endowed with the standard product and norm (u, v) H = ( u v + uv)dx; u 2 H = ( u 2 + u 2 )dx. D,2 ( ) is the completion of C0 ( ) with respect to the norm u 2 D := u 2 D,2 ( ) = u 2 dx. For any s + and Ω, L s (Ω) denotes a Lebesgue space; the norm in L s (Ω) is denoted by u s,ω, where Ω is a proper subset of, by s when Ω =. S is the best Sobolev constant for the Sobolev embedding D,2 ( ) L 6 ( ), that is, S = u D inf. u H ( )\{0} u 6 For any r > 0 and z, B r (z) denotes the ball of radius r centered at z. The letters c 0, d i, C 0 will be used to denote various positive constants which may vary from line to line and are not essential to the problem. We denote weak convergence and by strong convergence. Also if we take a subsequence of a sequence {u n }, we shall denote it again {u n }. We use o() to denote any quantity which tends to zero when n. Now we state our main results: Theorem Assume that (V 0 ) (V ), (K), (g) and (f ) (f 4 ) are satisfied. If V (x) < 0 for some x, then for each k N, there exist λ k > k, b k > 0 and η k > 0 such that problem (SP ) λ has at least two nontrivial solutions for every λ = λ k, g 2 η k and K 2 < b k ( or K < b k ). Theorem 2 Assume that (V 0 ) (V ), (K), (g) and (f ) (f 4 ) are satisfied. If V (0)

4 4 LIXIA WANG, SHIWANG MA AND NA XU has nonempty interior, then there exist Λ > 0, b λ > 0 and η λ > 0 such that problem (SP ) λ has at least two nontrivial solutions for every λ > Λ, g 2 η λ and K 2 < b λ ( or K < b λ ). If V 0, the restriction on the norm of K can be removed and we have the following theorem. Theorem Assume that V 0, (V 0 ) (V ), (K), (g) and (f ) (f 4 ) are satisfied. If V (0) has nonempty interior Ω, then there exist Λ > 0 and η > 0 such that problem (SP ) λ has at least two nontrivial solutions for every λ > Λ and g 2 η. To obtain our main results, we have to overcome several difficulties in using variational method. The main difficulty consists in the lack of compactness of the Sobolev embedding H ( ) into L p ( ), p (2, 6). Since we assume that the potential is no radially symmetric, we cannot use the usual way to recover compactness, for example, restricting in the subspace H r ( ) of radially symmetric functions or using concentration compactness methods. To recover the compactness, we borrow some ideas used in [5, 5] and establish the parameter dependent compactness conditions. Let us point out that the adaptation of the ideas to the procedure of our problem is not trivial at all, because of the presence of the nonlocal term K(x)φu.. Remark 2 (a) It is not difficult to find out functions f satisfying (f ) (f 4 ), for example, ( ) f(x, t) = h(x)t 2ln( + t 2 ) + t2 + t 2, (x, t) R, where h is a is a continuous bounded function with inf x h(x) > 0. (b) To the best of our knowledge, it seems that our theorems are the first results about the existence of multiple solutions for the nonhomogeneous Schrödinger- Poisson equations on with sign-changing potential and general nonlinear term. Although the methods are used before, we need to study carefully some properties of the term K(x)φu and the effect of the sign-changing potential V. The paper is organized as follows. In Section 2, we will introduce the variational setting for the problem and establish the compactness conditions. In Section, we give the proofs of main results. 2. Variational setting and compactness condition In this section, we give the variational setting of the problem (SP ) λ and establish the compactness conditions. Let V (x) = V + (x) V (x), where V ± = max{±v (x), 0}. Let { } E = u H ( ) : V + (x)u 2 dx < be equipped with the inner product and norm (u, v) = ( u v + V + (x)uv)dx, u = (u, u) /2.

5 SCHRÖDINGER-POISSON SYSTEMS 5 For λ > 0, we also need the following inner product and norm (u, v) λ = ( u v + λv + (x)uv)dx, u λ = (u, u) /2 λ. It is clear u u λ for λ. Set E λ = (E, λ ). It follows from (V 0 ) (V ) and the Poincaré inequality that the embedding E λ H ( ) is continuous, and hence, for s [2, 6], there exists d s > 0( independent of λ ) such that (2.) Let u s d s u λ, u E λ. F λ = {u E λ : suppu V ([0, ))}, and F λ denote the orthogonal complement of F λ in E λ. Clearly, F λ = E λ if V 0, otherwise F λ {0}. Define A λ := + λv, then A λ is formally self-adjoint in L 2 ( ) and the associated bilinear form a λ (u, v) = ( u v + λv (x)uv)dx is continuous in E λ. As in [5], for fixed λ > 0, we consider the eigenvalue problem (2.2) u + λv + (x)u = µλv (x)u, u F λ. Since (V 0 ) (V ), we see that the quadratic form u λv (x)u 2 dx is weakly continuous. Hence following Theorem 4.45 and Theorem 4.46 in [4], we deduce the following proposition, which is the spectral theorem for compact self-adjoint operators jointly with the Courant-Fischer minimax characterization of eigenvalues. Proposition 2. Assume that (V 0 ) (V ) hold, then for any fixed λ > 0, problem (2.2) has a sequence of positive eigenvalues {µ j (λ)}, which may be characterized by { } µ j (λ) = inf sup u 2 dimm j,m Fλ λ : u M, λv (x)u 2 dx =, j =, 2,,... Furthermore, µ (λ) µ 2 (λ) µ j (λ) + as j +, and the corresponding eigenfunctions {e j (λ)}, which may be be chosen so that (e i (λ), e j (λ)) λ = δ ij, are a basis of F λ. Now we give the properties for the eigenvalues {µ j (λ)} defined above. Proposition 2.2 ([5]) Assume that (V 0 ) (V ) hold and V 0. Then, for each fixed j N, (i) µ j (λ) 0 as λ +. (ii) µ j (λ) is a non-increasing continuous function of λ. Remark By Proposition 2.2 (i), there exists Λ 0 all λ > Λ 0. > 0 such that µ (λ) for Let E λ := span{e j(λ) : µ j (λ) } and E + λ := span{e j(λ) : µ j (λ) > }.

6 6 LIXIA WANG, SHIWANG MA AND NA XU Then E λ = E λ E + λ Fλ is an orthogonal decomposition. The quadratic form a λ is negative semidefinite on E λ, positive definite on E+ λ Fλ and it is easy to see that a λ (u, v) = 0 if u, v are in different subspaces of the above decomposition of E λ. From Remark, we have that dim E λ when λ > Λ 0. Moreover, since µ j (λ) + as j +, dim E λ < + for every fixed λ > 0. It is well known that problem (SP ) λ can be reduced to a single equation with a nonlocal term (see [2]). In fact, for every u E λ, the Lax-Milgram theorem implies that there exists a unique φ u D,2 ( ) such that (2.) with φ u = K(x)u 2 φ u (x) = 4π K(y)u 2 (y) dy. x y If K L ( ), by (2.), the Höder inequality and the Sobolev inequality, we get φ u 2 D = K(x)φ u u 2 dx S 2 d 4 2/5 K 2 u 4 λ. Similarly, if K L 2 ( ), φ u 2 D = K(x)φ u u 2 dx S 2 d 4 6 K 2 2 u 4 λ. Thus, there exists C 0 > 0, such that (2.4) φ u 2 D = K(x)φ u u 2 dx C 0 u 4 λ, K L 2 ( ) L ( ). Therefore, problem (SP ) λ can be reduced to the following equation: u + λv (x)u + K(x)φ u (x)u = f(x, u) + g(x), x. (SP ) λ is variational and its solutions are the critical points of the functional defined in E λ by I λ (u) = ( u 2 + λv (x)u 2 )dx + K(x)φ u u 2 dx F (x, u)dx 2 R 4 R g(x)udx. Furthermore, it is easy to prove that the functional I λ is of class C in E λ and that I λ(u), v = ( u v + λv (x)uv + K(x)φ u uv f(x, u)v g(x)v)dx for all u, v E λ. Hence, if u E λ is a critical point of I λ, then (u, φ u ) E λ D,2 ( ) is a solution of problem (SP ) λ. We refer the readers to [6] and [] for the details. Set N(u) = K(x)φ u u 2 dx = R 4π K(x)K(y)u 2 (x)u 2 (y) dxdy. R x y

7 SCHRÖDINGER-POISSON SYSTEMS 7 Now we give some properties about the functional N and its derivative N possess BL-splitting property, which is similar to Brezis-Lieb Lemma [7]. Proposition 2. ([8], Lemma 2.) Let K L ( ) L 2 ( ). H ( ) and u n (x) u(x) a.e. x, then (i) φ un φ u in D,2 ( ) and N(u) lim inf n N(u n ); (ii) N(u n u) = N(u n ) N(u) + o(); (iii) N (u n u) = N (u n ) N (u) + o() in H ( ). If u n u in Next, we investigate the compactness conditions for the functional I λ. Recall that a C functional J satisfies Cerami condition at level c ((C) c condition for short) if any sequence {u n } E such that J(u n ) c and ( + u n )J (u n ) 0 has a convergent subsequence; and such sequence is called a (C) c sequence. We only consider the case K L ( ), the other case K L 2 ( ) is similar. Lemma 2. Suppose that (V 0 ) (V ), (K), (f ) (f 4 ) and (g) are satisfied. Then every (C) c sequence of I λ is bounded in E λ for each c R. Proof. Let {u n } E λ be a (C) c sequence of I λ. Arguing indirectly, we can assume that (2.5) I λ (u n ) c, ( + u n λ )I λ(u n ) 0, u n λ as n after passing to a subsequence. Take w n := u n / u n λ. Then w n λ =, w n w in E λ and w n (x) w(x) a.e. x. We first consider the case w = 0. Combining this with (2.5), (f ) and the fact w n 0 in L 2 ({x : V (x) < 0}), we have o() = u n 2 λ ( I λ (u n ) ) 4 I λ(u n ), u n V (x)wndx 2 + R u n 2 λ wndx 2 suppv 4 g 2d 2 u n λ 4 w n 2 λ λ 4 4 λ 4 V = 4 + o(), F(x, u n )dx R 4 u n 2 g(x)u n dx λ a contradiction. If w 0, then the set Ω = {x : w(x) 0} has positive Lebesgue measure. For x Ω, one has u n (x) as n, and then, by (f 2 ), F (x, u n (x)) u 4 wn(x) 4 + as n, n(x) which, jointly with Fatou s lemma, shows that F (x, u n ) (2.6) Ω u 4 wndx 4 + as n. n We see from (f ), (2.4), the first limit of (2.5), (2.6) and (g) that C 0 F (x, u n ) F (x, u n ) lim sup 4 n R u n 4 dx lim sup λ n Ω u 4 wndx 4 = +. n

8 8 LIXIA WANG, SHIWANG MA AND NA XU This is impossible. Hence {u n } is bounded in E λ. Lemma 2.2 Suppose that (V 0 ), (V ), (K), (g) and (.2) hold. E λ, u n (x) u(x) a.e. in, and we denote w n := u n u, then (2.7) I λ (u n ) = I λ (w n ) + I λ (u) + o() If u n u in and (2.8) I λ(u n ), ϕ = I λ(w n ), ϕ + I λ(u), ϕ gϕdx + o(), for all ϕ E λ as n. In particular, if I λ (u n ) d and I λ (u n) 0 in Eλ ( the dual space of E λ ), then I λ (u) = 0 and (2.9) I λ (w n ) d I λ (u), I λ(w n ), ϕ gϕdx, for all ϕ E λ after passing to a subsequence. Proof. Since u n u in E λ, we have (u n u, u) λ 0 as n, which implies that (2.0) u n 2 λ = (w n + u, w n + u) λ = w n 2 λ + u 2 λ + o(). By (V 0 ), w n 0 and the Hölder inequality, we have V (x)w n udx = /2 V R suppv w n udx V wndx) (suppv 2 u 2 0 as n. Thus V (x)u 2 ndx = V (x)wndx 2 + V (x)u 2 dx + o(). Consequently, this together with Proposition 2. (ii) and (2.0), we obtain 2 a λ(u n, u n )+ ( 4 N(u n) = 2 a λ(w n, w n ) + ) ( 4 N(w n) + 2 a λ(u, u) + ) 4 N(u) +o(). Similarly, by Proposition 2. (iii), we have ( ) a λ (u n, h) + K(x)φ un u n hdx = a λ (w n, h) + K(x)φ wn w n hdx R ( ) + a λ (u, h) + K(x)φ u uhdx + o(), h E λ. Since g(x)u n dx = g(x)w n dx + g(x)udx, therefore, to obtain (2.7) and (2.8), it suffices to check that (2.) (F (x, u n ) F (x, w n ) F (x, u))dx = o() and (2.2) sup h λ = (f(x, u n ) f(x, w n ) f(x, u))hdx = o().

9 SCHRÖDINGER-POISSON SYSTEMS 9 Here, we only prove (2.), the verification of (2.2) is similar. Inspired by [2], we observe that ( ) d F (x, u n ) F (x, u n u) = dt F (x, u n tu) dt = f(x, u n tu)udt. and hence, by (.2), we have 0 F (x, u n ) F (x, u n u) ε u n u + ε u 2 + C ε u n p u + C ε u p, where ε, C ε > 0 and p (4, 6). Hence, for each ε > 0, and the Young inequality, we have F (x, u n ) F (x, w n ) F (x, u) C[ε u n 2 + C ε u 2 + ε u n p + C ε u p ]. Next, we consider the function g n given by g n (x) := max { F (x, u n ) F (x, w n ) F (x, u) Cε( u n 2 + u n p ), 0 }. Then 0 g n (x) CC ε ( u 2 + u p ) L ( ). Moreover, by the Lebesgue dominated convergence theorem, (2.) g n (x)dx 0 as n, since u n u a.e. in. By the definition of g n, it follows that F (x, u n ) F (x, w n ) F (x, u) g n (x) + Cε( u n 2 + u n p ), which, together with (2.) and (2.) shows that F (x, u n ) F (x, w n ) F (x, u) Cε for n sufficiently large. Which implies that [F (x, u n ) F (x, w n ) F (x, u)]dx = o(). Next, we check that I λ (u) = 0. Indeed, for each ψ C 0 ( ), we have (2.4) and (2.5) ( V (x)(u n u)ψdx V (u n u, ψ) λ 0 as n. suppψ 0 (u n u) 2 dx) /2 ψ 2 0 as n, since u n u in L 2 loc (R ). By Proposition 2. (i), u n u in E λ yields φ un φ u in D,2 ( ). So φ un φ u in L 6 ( ). For every ψ C0 ( ), by the Hölder inequality we obtain K(x)uψ 6/5 dx K 6/5 ψ 6/5 2/5 u 6/5 2/5, that is K(x)uψ L 6/5 ( ), and hence K(x)(φ un φ u )uψdx 0.

10 0 LIXIA WANG, SHIWANG MA AND NA XU By u n u in L loc (R ) and the Hölder inequality, we have K(x)φ un (u n u)ψdx ψ 2 K φ un 6 u n u,ωψ C u n u,ωψ 0 as n, here Ω ψ is the support set of ψ. Consequently, [K(x)φ un u n ψ K(x)φ u uψ]dx (2.6) K(x)φ un (u n u)ψ dx + K(x)(φ un φ u )uψ dx = o(). Furthermore, by (.2) and the dominated convergence theorem, we obtain [f(x, u n ) f(x, u)]ψdx = [f(x, u n ) f(x, u)]ψdx = o(). Ω ψ Since u n u in L 2 ( ) and g L 2 ( ), we obtain g(u n u)dx = o(). This jointly with (2.4), (2.5), (2.6) and the dominated convergence theorem, shows that I λ(u), ψ = lim n I λ(u n ), ψ = 0, ψ C 0 ( ). Hence I λ (u) = 0. (2.9) follows from (2.7)-(2.8) and Proposition 2.(iii). The proof is complete. Lemma 2. Suppose V 0, (V 0 ), (V ), (K), (g) and (f ) (f 4 ) hold. Then, for any M > 0, there is Λ = Λ(M) > 0 such that I λ satisfies (C) c condition for all c < M and λ > Λ. Proof. Let {u n } E λ be a (C) c sequence with c < M. According to Lemma 2., {u n } is bounded in E λ, and there exists C λ such that u n λ C λ. Therefore, up to a subsequence, we can assume that (2.7) u n u in E λ ; u n u in L s loc( )(2 s < 2 ); u n (x) u(x) a.e. x. Now we can prove that u n u in E λ for λ > 0 large. Denote w n := u n u, then w n 0 in E λ. By Lemma 2.2, we have I λ (u) = 0, and (2.8) I λ (w n ) c I λ (u), I λ(w n ) 0 as n. Noting V 0 and using (f ), we have I λ (u) = I λ (u) 4 I λ(u), u = 4 u 2 λ + F(x, u)dx gudx = Φ λ (u) gudx, R 4 R 4 here Φ λ (u) = 4 u 2 λ + F(x, u)dx 0.

11 SCHRÖDINGER-POISSON SYSTEMS Again by (2.9) and (2.7), we obtain 4 w n 2 λ + F(x, w n )dx + o() = I λ (w n ) R 4 I λ(w n ), w n = c I λ (u) + gw n dx + o() 4 R [ = c Φ λ (u) ] gudx + (2.9) gw n dx + o() 4 R 4 = c Φ λ (u) + gudx + o() 4 M + M + o(). Here we use the fact c < M and 4 g 2 u 2 4 g 2d 2 u λ 4 g 2d 2 lim inf n u n λ g 2 d 2 C M, where M is a positive constant independent of λ. Hence (2.20) F(x, w n )dx M + M + o(). Since V (x) < b on a set of finite measure and w n 0, w n 2 2 (2.2) λv + (x)w 2 λb ndx + wndx 2 λb w n 2 λ + o(). V b For 2 < s < 2, by (2.2) and the Hölder and Sobolev inequality, we obtain ( ) 6 s ( ) s 2 w n s s = w n s dx w n 2 4 dx w n 6 4 dx [ ] 6 s ( [ ] ) s 2 ( wn 2 + λv + wn 2 ) 4 4 dx S 6 w n 2 dx + o() λb ( ) 6 s 4 (2.22) S (s 2) 2 w n s λ + o(). λb By (f ), for any ε > 0, there exists δ = δ(ε) > 0 such that f(x, t) ε t for all x and t δ, and (f 4 ) is satisfied for t δ (with the same τ but possibly larger a ). Hence we get (2.2) f(x, w n )w n dx ε wndx 2 ε λb w n 2 λ + o(), and (2.24) w n δ w n δ f(x, w n )w n dx ( ( w n δ w n δ V <b ) /τ f(x, w n) τ dx w n 2 s w n ) /τ w n 2 a F(x, w n )dx w n δ [a (M + M)] /τ S (2s 4) 2s s ( ) θ w n 2 λ + o() λb

12 2 LIXIA WANG, SHIWANG MA AND NA XU by (f 4 ), (2.20), (2.22) with s = 2τ/(τ ) and the Hölder inequality, where θ = 6 s 2s > 0. Since u n u in L 2 ( ) and g L 2 ( ), we have (2.25) g(u n u)dx 0 as n. Therefore, by (2.2), (2.24) and (2.25) we have o() = I λ(w n ), w n w n 2 λ f(x, w n )w n dx gw n dx R [ ε ( ) ] θ λb [a (M + M)] /τ S (2s 4) (2.26) 2s w n 2 λ + o(). λb So, there exists Λ = Λ(M) > 0 such that w n 0 in E λ when λ > Λ. Since w n = u n u, it follows that u n u in E λ. This completes the proof. Lemma 2.4 Let (V 0 ), (V ), (K), (g) and (f ) (f 4 ) be satisfied. Let {u n } be a (C) c sequence of I λ with level c > 0. Then for any M > 0, there is Λ = Λ(M) > 0 such that, up to a subsequence, u n u in E λ with u being a nontrivial critical point of I λ and satisfying I λ (u) c for all c < M and λ > Λ. Proof. We modify the proof of Lemma 2.. By Lemma 2.2, we obtain (2.27) I λ(u) = 0, I λ (w n ) c I λ (u), I λ(w n ) 0 as n. However, since V is allowed to be sign-changing and the appearance of nonlinear term g, from I λ (u) = I λ (u) 4 I λ(u), u = 4 u 2 λ λ V (x)u 2 dx + F(x, u)dx gudx, 4 R 4 we cannot deduce that I λ (u) 0. We consider two possibilities: (i) I λ (u) < 0; (ii) I λ (u) 0. If I λ (u) < 0, then u 0 and then u is nontrivial and the proof is done. If I λ (u) 0, following the argument in the proof of Lemma 2. step by step, we can get u n u in E λ. In fact, by (V 0 ) and w n 0 in L 2 ({x : V (x) < b}), we have V (x)wn(x)dx 2 V wndx R suppv 2 = o(). Combining this with (2.27), we have F(x, w n )dx = I λ (w n ) 4 I λ(w n ), w n 4 w n 2 λ + 4 c I λ (u) + o() M + o(). λv (x)w 2 ndx + 4 gw n dx

13 SCHRÖDINGER-POISSON SYSTEMS It follows that (2.24), (2.25) and (2.26) remain valid. Hence u n u in E λ and I λ (u) = c(> 0). The proof is complete.. Proofs of main results If V is sign-changing, we first verify that the functional I λ have the linking geometry to apply the following linking theorem [22]. Proposition. Let E = E E2 be a Banach space with dim E 2 <, Φ C (E, ). If there exist R > ρ > 0, α > 0 and e 0 E such that α := inf Φ(E S ρ ) > sup Φ( Q) where S ρ = {u E : u = ρ}, Q = {u = v + te 0 : v E 2, t 0, u R}. Then Φ has a (C) c sequence with c [α, sup Φ(Q)]. In our context, we use Proposition. with E = E + λ Fλ and E 2 = E λ. By Proposition 2.2, µ j (λ) 0 as λ for every fixed j. By Remark, there is Λ 0 > 0 such that E λ and E λ is finite dimensional for λ > Λ 0. Now we investigate the linking structure of the functional I λ. Lemma. Suppose that (V 0 ), (V ), (K), (g) and (.2) with p (4, 2 ) are satisfied. Then, for each λ > Λ 0 ( is the constant given in Remark 2), there exist α λ, ρ λ and η λ > 0 such that (.) I λ (u) α λ for all u E + λ Fλ with u λ = ρ λ and g 2 < η λ. Furthermore, if V 0, we can choose α, ρ, η > 0 independent of λ. Proof. For any u E + λ Fλ, writing u = u + u 2 with u E + λ and u 2 F λ. Clearly, (u, u 2 ) λ = 0, and (.2) ( u 2 + λv (x)u 2 )dx = ( u 2 + λv (x)u 2 )dx + u 2 2 λ. By Proposition 2. we have µ j (λ) + as j + for each fixed λ > Λ 0. So there is a positive integer n λ such that µ j (λ) for j n λ and µ j (λ) > for j > n λ +. For u E + λ, we set u = Σ j=n λ + µ j(λ)e j (λ). Thus (.) ( u 2 + λv (x)u 2 )dx = u 2 λ λv (x)u 2 dx Now, using (2.), (.2) and (.), we obtain ( ) u 2 µ λ. nλ +(λ) I λ (u) ( ) u 2 λ ε u 2 2 C ε u p p g 2 u 2 2 µ nλ +(λ) ( ) u 2 λ εd 2 2 µ 2 u 2 λ C ε d p p u p λ nλ +(λ) d 2 g 2 u λ {[ ( ) ] } u λ εd 2 2 u λ C ε d p 2 µ p u p λ d 2 g 2. nλ +(λ)

14 4 LIXIA WANG, SHIWANG MA AND NA XU Let h(t) = ρ(λ) = [ ] 2 ( µ nλ +(λ) εd2 2 t C ε d p pt p, for t > 0, p (4, 6) there exists [ 2 ( µ nλ + (λ) ) εd2 2 C εd p p(p ) ] p 2 such that max t 0 h(t) = h(ρ(λ)) > 0. It follows from above inequality, I λ (u) u λ =ρ(λ)> 0 for all g 2 < η λ := h(ρ(λ)) 2d 2. Of course, ρ(λ) can be chosen small enough, we can obtain the same result: there exists α λ > 0, such that I λ (u) α λ, here u λ = ρ λ. If V 0, since E λ = F λ, and ( u 2 + λv (x)u 2 )dx = u 2 λ, we can choose α, ρ, η > 0 (independent of λ) such that (.) holds. Lemma.2 Suppose that (V 0 ), (V ), (K), (g) and (f ) (f 2 ) are satisfied. for any finite dimensional subspace Ẽλ E λ, there holds I λ (u) as u λ, u Ẽλ. Then, Proof. Assuming the contrary, there is a sequence (u n ) Ẽλ with u n λ such that (.4) < inf n I λ(u n ). Take v n := u n / u n λ. Since dim Ẽλ < +, there exists v Ẽλ \ {0} such that v n v in Ẽ λ, v n (x) v(x) a.e. x after passing to a subsequence. If v(x) 0, then u n (x) + as n, and hence by (f 2 ), F (x, u n (x)) u 4 vn(x) 4 + as n. n(x) Combining this with (f ), (2.4) and Fatou s lemma, we have I λ (u n ) u n 4 λ 2 u n 2 + C 0 λ 4 F (x, u n ) u n R u n 4 dx g(x) λ R u n 4 dx λ 2 u n 2 + C ( ) 0 F (x, λ 4 un ) + v=0 v 0 u 4 vndx 4 + g 2d 2 n u n λ 2 u n 2 + C 0 λ 4 F (x, u n ) v 0 u 4 vndx 4 + g 2d 2 n u n λ, a contradiction with (.4). Lemma. Suppose that (V 0 ), (V ), (K), (g) and (f ) (f 2 ) are satisfied. If V (x) < 0 for some x, then, for each k N, there exist λ k > k, w k E + λ Fλk k, R λk > ρ λk (ρ λk is the constant given in Lemma.) and η k, b k > 0 such that, for g 2 < η k and K < b k ( or K 2 < b k ), (a) sup I λk ( Q k ) 0, (b) sup I λk (Q k ) is bounded above by a constant independent of λ k, where Q k := {u = v + tw k : v E λ k, t 0, u λk R λk }.

15 SCHRÖDINGER-POISSON SYSTEMS 5 Proof. We adapt an argument in Ding and Szulkin [5]. For each k N, since µ j (k) + as j, there exists j k N such that µ jk (k) >. By Proposition 2.2, there is λ k > k such that < µ jk (λ k ) < + λ k. Taking w k := e jk (λ k ) be an eigenvalue of µ jk (λ k ), then w k E + λ k as µ jk (λ k ) >. Since dim E λ Rwk k < +, it follows directly from Lemma.2 that (a) holds with R λk > 0 large enough. By (f 2 ), for each η > V, there is r η > 0 such that F (x, t) 2 ηt2 if t r η. For u = v + w E λ Rwk k, we obtain V (x)u 2 dx = V (x)v 2 dx + V (x)w 2 dx by the orthogonality of E λ k and Rw k. Hence we get I λk (u) ( w 2 + λ k V (x)w 2) dx + 2 R 4 gudx 2 [µ j k (λ k ) ]λ k K(x)φ u u 2 dx F (x, u)dx suppv V (x)w 2 dx ηu 2 dx+ R 2 suppv 4 S 2 d 4 2/5 K 2 u 4 λ k + d 2 g 2 u λk (F (x, u) 2 ) ηu2 dx suppv, u r η V (x)w 2 η dx 2 R 2 V V (x)w 2 dx + C η + R 4 S 2 d 4 2/5 K 2 Rλ 4 k + d 2 g 2 R λk C η + for u = v + w E λ Rwk k with u λk R λk, K < b k := S(d 2/5 R λk ) 2 and g 2 < η k := 2d 2R λk, where C η depends on η but not λ. Lemma.4 Suppose that (V 0 ), (V ), (K), (g) and (f ) (f 2 ) are satisfied. If Ω := intv (0) is nonempty, then, for each λ > Λ 0 ( is the constant given in Remark 2), there exist w E + λ Fλ, R λ > 0, η λ > 0 and b λ > 0 such that for g 2 < η λ, K < b λ ( or K 2 < b λ ), (a) sup I λ ( Q) 0, (b) sup I λ (Q) is bounded above by a constant independent of λ, where Q := {u = v + tw : v E λ, t 0, u λ R λ }. Proof. Choose e 0 C 0 (Ω) \ {0}, then e 0 F λ. By Lemma.2, there is R λ > 0 large such that I λ (u) 0 where u E λ Re0 and u λ R λ.

16 6 LIXIA WANG, SHIWANG MA AND NA XU For u = v + w E λ Re0, we have I λ (u) w 2 dx + K(x)φ u u 2 dx 2 R 4 w 2 dx η u 2 dx 2 R 2 Ω (.5) Ω, u r η Ω F (x, u)dx gudx (F (x, u) η2 u2 ) dx + 4 S 2 d 4 2/5 K 2 u 4 λ + g 2 d 2 u λ w 2 dx η u 2 dx + C η + 2 R 2 Ω 4 S 2 d 4 2/5 K 2 u 4 λ + g 2 d 2 u λ. Observing w C 0 (Ω), we have (.6) w 2 dx = ( w)udx w 2 u 2,Ω c 0 w 2 u 2,Ω c2 0 Ω 2 η w η 2 u 2 2,Ω, where c 0 is a constant depending on e 0. Choosing η > c 2 0, we have w 2 2 η u 2 2,Ω, and it follows from (.5) that I λ (u) C η + 4 S 2 d 4 2/5 K 2 R 4 λ + g 2 d 2 R λ C η + for all u E λ Re0 with u λ R λ and K < b λ := S(d 2/5 R λ ) 2 and g 2 < η λ := 2d 2R λ, where C η depends on η but not λ. Now we are in a position to prove our main results. Proof of Theorem. The proof of this theorem is divided in two steps. Step There exists a function u λ E λ such that I λ (u λ) = 0 and I λ (u λ ) < 0. Since g L 2 ( ) and g 0( 0), we can choose a function ψ E λ such that g(x)ψ(x)dx > 0. Hence, we have I(tψ) = t2 2 ψ 2 λ λt2 V (x)ψ 2 dx + t4 K(x)φ ψ ψ 2 dx 2 R 4 R F (x, tψ)dx t g(x)ψdx t2 2 ψ 2 λ + t4 4 C 0 ψ 4 λ t g(x)ψdx < 0 for t > 0 small enough. Thus, there exists u λ small enough such that I λ (u λ ) < 0. By Lemma., we have c 0,λ = inf{i λ (u) : u B ρλ } < 0, where ρ λ > 0 is given by Lemma., B ρλ = {u E λ : u λ < ρ λ }. By the Ekeland s variational principle, there exists a minimizing sequence {u n,λ } B ρλ such that c 0,λ I λ (u n,λ ) < c 0,λ + n λ,

17 SCHRÖDINGER-POISSON SYSTEMS 7 and I λ (w λ ) I λ (u n,λ ) n λ w λ u n,λ λ for all w λ B ρλ. Clearly, {u n,λ } is a bounded Palais-Smale sequence of I λ. Then, by a standard procedure, Lemma 2. and Lemma 2.2 imply that there exists a function u λ E λ such that I λ (u λ) = 0 and I λ (u λ ) = c 0,λ < 0. If V 0, we can get ρ λ, c 0,λ, u 0,λ are independent of λ. Step 2 There exists a function ũ λ E λ such that I λ (ũ λ) = 0 and I λ (ũ λ ) > 0. It follows from Lemmas.,. and Proposition. that, for each k N, λ = λ k, g 2 < η k and 0 < K < b k (or 0 < K 2 < b k ), I λk has a (C)c sequence with c [α λk, sup I λk (Q k )]. Setting M := sup I λk (Q k ), then I λk has a nontrivial critical point according to Lemmas 2., 2.4 and Proposition.. That is, there exists a function ũ λ E λ such that I λ (ũ λ) = 0 and I λ (ũ λ ) = c α λk > 0. The proof is complete. Proof of Theorem 2. The first solution is similar to the first solution of Theorem. The second solution follows from Lemmas 2., 2.4,.,.4 and Proposition.. The proof is complete. Proof of Theorem. The proof of this theorem is divided in two steps. Step There exists a function u 0 E λ such that I λ (u 0) = 0 and I λ (u 0 ) < 0. In the proof of Theorem, we can choose c 0 = c 0,λ, B ρ = B ρ,λ, then by the By the Ekeland s variational principle, there exists a sequence {u n } B ρ such that and c 0 I λ (u n ) < c 0 + n, I λ (w) I λ (u n ) n w u n λ for all w B ρ. Then by a standard procedure, we can show that {u n } is a bounded Palais-Smale sequence of I λ. Therefore Lemmas 2. and 2.2 imply that there exists a function u 0 E λ such that I λ (u 0) = 0 and I λ (u 0 ) = c 0 < 0. Step 2 There exists a function ũ λ E λ such that I λ (ũ λ) = 0 and I λ (ũ λ ) > 0. Since we suppose V 0, the functional I λ has mountain pass geometry and the existence of nontrivial solutions can be obtained by mountain pass theorem [22]. In fact, by Lemma., there exist constants α, ρ, η > 0 ( independent of λ) such that, for each λ > Λ 0, I λ (u) α for u E λ with u λ = ρ and g 2 < η. Take e C0 (Ω) \ {0}, by (f ), (f 2 ) and Fatou s lemma, we get I λ (te) t 4 2t 2 e 2 dx + 4 N(e) F (x, te) (te) 4 e 4 dx t Ω {x Ω:e(x) 0} Ω gedx

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