Existence of solutions for a class of Kirchhoff-type equations with indefinite potential

In this paper, we consider the existence of solutions of the following Kirchhoff-type problem: {−(a+b∫R3|∇u|2dx)Δu+V(x)u=f(x,u),in R3,u∈H1(R3),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx )\Delta u+ V(x)u=f(x,u) , & \text{in }\mathbb{R}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}),\end{cases}\displaystyle \end{aligned}$$ \end{document} where a,b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a,b>0$\end{document} are constants, and the potential V(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V(x)$\end{document} is indefinite in sign. Under some suitable assumptions on f, the existence of solutions is obtained by Morse theory.


Introduction and main result
In this paper, we consider the existence of solutions of the following Kirchhoff-type problem where a, b are postive constants, and the potential V(x) is continuous and indefinite in sign.The nonlinear term R 3 |∇u| 2 dx appears in (1.1), which means that (1.1) is not a pointwise identity.This leads to some mathematical difficulties that make the research particularly interesting.(1.1) has an interesting physics background.When V(x) = 0, and a bounded domain Ω ⊂ R N is substituted R 3 , then we obtain the following nonlocal Kirchhoff-type problem 2) The problem (1.2) is regard to the stationary analogue of the equation which was presented by Kirchhoff in [1], and (1.3) is a generalization of the classical D'Alembert's wave equation for free vibrations of elastic strings.Problem (1.3) has been increasingly more attention after Lions in [2] introduced an abstract framework to the problem.We can refer to [3,4,5] for the physical and mathematical background of this problem.
In this paper, we will consider V(x) is indefinite in sign and do not assume any compactness condition on V(x) which is different from most of the articles mentioned above.Motived by Chen [24] and Sun [25], we overcome two difficulties, namely, verifying the link geometry and the boundedness of Cerami sequence for the corresponding functional of (1.1).We obtain the existence of solutions for (1.1) by the Symmetric Montain Pass Theorem. Set . Before stating our main result, we make the following assumptions: and V(x) is bounded from below, and there is M > 0 such that the set x ∈ R 3 |V + (x) < M is nonempty and has finite measure.(V 2 ) There exists a constant η 0 > 1 such that , and there exist constants p ∈ (2, 6) and c > 0 such that 2 Now, we are ready to state the main result of this paper: possesses infinitely many solutions.

Preliminaries
We work in the Hilbert space with the inner product and the norm , ∀u ∈ E.
The problem (1.1) has a variational structure, then a weak solution of problem (1.1) is a critical point of the following functional Then under the assumptions (V 1 ), ( f 1 ) and ( f 2 ), the functional Φ ∈ C 1 (E, R) and for all u, v ∈ E, For any s ∈ [2,6], since the embedding E ֒→ L s (R 3 ) is continuous, there exists a constant (2.4) To complete the proof of theorem 1.1, we need the following Symmetric Mountain Pass Theorem: Theorem 2.1.( [26]) Let X be an infinite demensional Banach space, X = Y ⊕ Z, where Y is finite dimensional.If I ∈ C 1 (X, R) satisfies (C) c -condition for all c > 0, and (I 1 ) I(0) = 0, I(−u) = I(u), ∀u ∈ X; (I 2 ) there exist constants α, ρ > 0, such that I| ∂B ρ ∩Z ≥ α; (I 3 ) for any finite dimensional subspace X ⊂ X, there is R = R( X) > 0, such that I(u) ≤ 0 on X \ B R ; then I possesses an unbounded sequence of critical values.
Definition 2.2.Assume E be a Banach space, and We say that Φ satisfies the Cerami condition at level c (shortly, (C) c -condition) if every (C) c sequence of Φ contains a convergent subsequence.If Φ satisfies (C) c -condition for every c ∈ R, then we say that Φ satisfies the Cerami condition (shortly, (C)-condition ).
Proof.From Lemma 3.1 we know that any (C) c sequence {u n } is bounded in E.Then, passing to a subsequence, we may assume that ).Note that, by (2.2) Since u n ⇀ u in E, we know that R 3 ∇u∇(u n − u)dx → 0 as n → ∞.Consequently, by the boundedness of {u n } in E, we have b Noting that V − (x) ≥ 0 for all x ∈ R 3 and (V 1 ) implies that V − ∈ L ∞ (R 3 ).Moreover, it follows from (V 1 ) that {V + = 0} has finite measure, which implies that {V − (x) > 0} has finite measure.Since → 0, as n → ∞. (3.9) Next, let ε > 0, for l ≥ 1, it follows from ( f 1 ) and Hölder inequality that since p < 6, we may fix l large enough such that for all n.Moreover, by ( f 4 ) there exists L > 0 such that for all n.For any ε > 0, by ( f 1 ) and ( f 2 ), there exists and where 2 < p < 6.Since u n → u in L s (B L (0)) for s ∈ [2, 6), from (3.12) we have for n large enough.Combining (3.10), (3.11) (3.14), we conclude that for n large enough.Since ε is arbitrary, (3.15), together with (3.6)-(3.9),we get u n → u .Thus, u n → u in E.
Proof of Theorem 1.1 Let e j is a total orthonormal basis of E and define X j = Re j , Proof.Obviously, Φ(0) = 0 and Φ is even due to f is odd, we will verify that Φ satisfies the remain conditions of Theorem 2.1.Firstly, we can verify that Φ satisfies (I 2 ).By (2.4) and (3.13) with 0 < ε < η 0 −1 , we have for all u ∈ ∂B ρ , where B ρ = {u ∈ E : u < ρ}.Therefore, for ρ small enough.