On the Existence of Solutions for a Class of Schr¨odinger–Kirchhoff-Type Equations with Sign-Changing Potential

where a, 0 are constants, λ is a positive parameter, and 4 < p < 6. Under suitable assumptions on V ( x ) , the existence of nontrivial solution is obtained via variational methods. The potential V ( x ) is allowed to be


Introduction and Main Results
In this paper, we consider the following Kirchho type problem: where a, b > 0 are constants, λ is a positive parameter, 4 < p < 6, and the potential V satis es the following conditions: (V 1 )V ∈ C(R 3 , R) and V is bounded below (V 2 ) there exists a constant c > 0 such that the set x ∈ R 3 : V(x) ≤ c is nonempty and meas x ∈ R 3 : V(x) ≤ c < + ∞, where meas denote the Lebesgue measure in R 3 is kind of assumptions was rst introduced by Bartsch and Qiang Wang [1] in the study of the nonlinear Schrödinger equations and has attracted the attention of several researchers.
In recent years, the Kirchho problem on a bounded domain Ω ⊂ R N − a + b R 3 |∇u| 2 dx Δu + V(x)u |u| p− 2 u, in Ω, u 0, on zΩ , has been studied by many authors (see, for example, [2][3][4][5][6][7][8]). More recently, many researchers focused on the Kirchho problem de ned on the whole space R 3 , i.e., the following problem: where V: R 3 ⟶ R is a potential function and f ∈ C(R 3 × R, R). In [9], Wu studied (3) by using a symmetric Mountain Pass eorem under the following assumptions about potential V Under this condition, by Lemma 3.4 in [10], the embedding H 1 (R 3 )↪L s R 3 is compact for any s ∈ [2,6). Hence, the corresponding results in [9] have been obtained by using the variational techniques in a standard way. In [11][12][13], the authors considered Kirchhoff type problem (3) with a steep potential well. Precisely, the potential function satisfies the following conditions besides (V 2 ): is a nonempty open set with locally Lipschitz boundary and Ω � V − 1 (0) By using this conditions, Sun and Wu [11] considered (3) in the case where the nonlinearity f(x, s) is asymptotically k-linear (k � 1, 2, 4) with respect to s at infinity. Du et al. [12] studied (3) when f(x, u) behaves like |u| p− 2 u with 4 < p < 6 and proved the existence and asymptotic behavior of ground state solutions. Zhang and Du [13] investigated the existence and asymptotic behavior of positive solutions for (3) by combining the truncation technique and the parameter-dependent compactness lemma for b small and λ large in the case where f(x, u) behave like |u| p− 2 u with 2 < p < 4. For more results about Kirchhoff type problems, we refer the reader to [14][15][16][17][18] and the references therein.
Under the assumption of (V 1 ), the potential V may change sign. e purpose of this paper is to consider the multiplicity of solutions for (1) in this case. To our best knowledge, there is no existence result of solutions for (1) with sign-changing potentials. Our main result as follows.
en, system (1) possesses infinitely many distinct pairs of nontrivial solutions whenever λ > 0 is sufficiently large.

Preliminaries
As a matter of convenience, without loss of generality, we may assume that a � 1 and b � 1. Consequently, we are dealing with the Kirchhoff type problem as be the usual Sobolev space with the standard inner product and norm as follows: In our problem, we work in the space defined by with the inner product and the norm as follows: where It follows from the conditions (V 1 ) and (V 2 ) and the Hölder and Sobolev inequalities that which implies that the embedding Here, S is the best constant for the embedding of Combine with the continuity of the following embedding: ere is a constant a s > 0 such that As a consequence, the functional I λ : E λ ⟶ R given by is well defined, and it is of class C 1 with derivative 2 Discrete Dynamics in Nature and Society for all u, v ∈ E λ . As in [19], let and denote the orthogonal complement of F λ in E λ by F ⊥ λ . Consider the eigenvalue problem In view of (V 1 ) and (V 2 ), the quadratic form u↦ R 3 V − (x)u 2 dx is weakly continuous. We have the following proposition.
Theorem 2 (see Theorem 9.12 in [20]). Let E be an infinite dimensional Banach space, and let I ∈ C 1 (E, R) be even, satisfying (PS) condition and I(0) � 0. If E � V⊕X, where V is finite dimensional and I satisfies the following: then I possesses an unbounded sequence of critical values.

Proof of Main Results
Proof. By Proposition 1, for each fixed λ > Λ, there exists a positive integer n λ such that μ j (λ) ≤ 1 for j < n λ and μ j (λ) > 1 for j ≥ n λ . us, for any u � u 1 + u 2 ∈ E + λ ⊕F λ , we have for all u ∈ B ρ (0), where B ρ (0) � u ∈ E + λ ⊕F λ : ‖u‖ E λ < ρ . Since p > 2, the conclusion follows by choosing ρ sufficiently small. Proof. Since all norms are equivalent in a finite dimensional space, there are constants C p > 0 and C > 0 such that where ‖u‖ 2 Discrete Dynamics in Nature and Society Since p > 4, consequently, there is a large r > 0 such that I(u) < 0 on E/B r (0). Proof. Let u n be a (PS) c sequence, that is, I λ (u n ) ⟶ c and I λ ′ (u n ) ⟶ 0. If u n is unbounded in E λ , up to a subsequence, we can assume that as n ⟶ ∞, after passing to a subsequence. Set w n � u n /‖u n ‖ E λ , we can assume that w n ⇀w in E λ and a contradiction. If w ≠ 0, then the set Ω � x ∈ R 3 : ω(x) ≠ 0 has positive Lebesgue measure. For x ∈ Ω, one has |u n (x)| ⟶ ∞ as n ⟶ ∞; Fatou's lemma shows that Ω |u n | p− 4 w 4 n dx ⟶ ∞ as n ⟶ ∞. us, by (9), we obtain is is a contradiction. is implies u n is bounded in E λ . We assume that ‖u n ‖ E λ ≤ T. Passing to a subsequence if necessary, we can assume that there exists u ∈ E λ and A ∈ R such that 4 Discrete Dynamics in Nature and Society (24) en, I λ ′ (u n ) ⟶ 0 implies that Taking v � u in (25), we obtain Let v n ≔ u n − u. It follows from (V 1 ) and Moreover, Let 0 < α < min 6 − p/2, 1 , 2 < p < 6. en, 2 < 2(p − α)/2 − α < 6. By Sobolev inequalities and Hölder inequality, one has we know