Improved results of nontrivial solutions for a nonlinear nonhomogeneous Klein–Gordon–Maxwell system involving sign-changing potential

where λ ≥ 1 is a parameter, ω > 0 is a constant and the potential A is sign-changing. Under the classic Ambrosetti–Rabinowitz condition and other suitable conditions, nontrivial solutions are obtained via the linking theorem and Ekeland’s variational principle. Especially speaking, we use a super-quadratic condition to replace the 4-superlinear condition which is usually used to show the existence of nontrivial solutions in many references. Our results improve the previous results in the literature.


Introduction
The following type of Klein-Gordon-Maxwell system is considered: where λ ≥ 1 is a parameter, ω > 0 is a constant, A ∈ C(R 3 , R), f ∈ C(R 3 × R, R), and f satisfies the following basic condition: (F1) f (x, t) = o(|t|) uniformly in x as t → 0, there exists a constant C > 0 such that |f (x, t)| ≤ C(|t| + |t| q ), 2 < q < 6, for all (x, t), and F(x, t) = The Klein-Gordon-Maxwell system was first introduced by Benci and Fortunato [1] as a model to describe a nonlinear Klein-Gordon equation interacting with an electromagnetic field.
When the potential A was an external Coulomb function, or a steep function, or a periodic function, or a sign-changing function, etc., the Klein-Gordon-Maxwell system had been extensively studied in the past decades. For example, positive ground state solutions for the following system were obtained by Cunha [5]: where A is a periodic potential.
In [6], Georgiev and Visciglia investigated a homogeneous system with a small external Coulomb potential and λ = 1. In [7], Chen and Tang considered the geometrically distinct solutions for Klein-Gordon-Maxwell systems by using Lusternik-Schnirelmann theory. In [8], Ding and Li proved that the Klein-Gordon-Maxwell system with sign-changing potential had infinitely many standing wave solutions. Liu, Chen and Tang [9] studied the ground state solutions for Klein-Gordon-Maxwell system with steep potential well. In [10], Wang improved the results of [2].
Next, let us present some results for the nonhomogeneous case. When λ ≡ 1 and K(x) ≡ 1, Shi and Chen [11] established the multiplicity of solutions for nonhomogeneous system (1.1). Wang and Chen [12] investigated the system (1.1) with sign-changing potential A and f satisfies the following crucial assumptions: Existence and multiplicity of solutions for a type of Klein-Gordon-Maxwell system with sign-changing potentials were got via the symmetric mountain pass theorem in [13]. In [14], under a variant super-quadratic condition, two solutions for a nonhomogeneous Klein-Gordon-Maxwell system were got by Wang via the mountain pass theorem and Ekeland's variational principle. Via Ekeland's variational principle and the mountain pass theorem, the author in [15] studied the nonhomogeneous Klein-Gordon-Maxwell system with constant potential.
Finally, we mention some recent work also related to the Klein-Gordon-Maxwell system. In [16], the authors investigated positive ground state solutions for a kind of fractional Klein-Gordon-Maxwell system. In [17,18], some results on reaction-diffusion equations involving fractional operators were obtained. In [19], some results on Klein-Gordon equations involving fractional operators were obtained. The authors in [20] studied a nonlinear heat equation and obtained some results. In [21], the authors investigated the numerical computation of Klein-Gordon equation by using a homotopy analysis transform method.
Inspired by the above-mentioned work, we will make the following conditions which are weaker than conditions (F2)' and (F3)' . Otherwise, we consider a more general potential . In our assumptions, the nonlinearity f just needs to satisfy a super-quadratic condition at infinity. The 4-superlinear assumption is not necessary. Conditions (A1)-(A3) were first introduced in [22]. Since the potential in (1.1) is sign-changing, the usual way of verifying the compactness is invalid. Following [12,23], we establish the parameter which is dependent on compactness conditions to recover the compactness. The following assumptions will be needed throughout the paper.
(F4) There exist a 1 , L 0 > 0 and σ ∈ (3/2, 2) such that where ∈ (2, 4), c is a continuous function with inf x∈R 3 c(x) > 0. If there exists x 0 ∈ R 3 such that A(x 0 ) < 0, then ∀n ∈ N, there exist λ n > n, b n > 0 and η n > 0 such that system (1.1) admits at least two nontrivial solutions for every λ = λ n , |K| ∞ < b n (or |K| 3 < b n ) and |h| 2 ≤ η n .  In the present paper, we use weaker conditions than the previous literature to show the boundedness of Palais-Smale sequence and obtain nontrivial solutions for the nonhomogeneous Klein-Gordon-Maxwell system involving sign-changing potential, which extend and generalize the related results in the literature.
In this paper, C denotes different positive constant in different place. The paper is organized by four sections. Some preliminary results are stated in Sect. 2. The proofs of main results are given in Sect. 3. The conclusion is given in Sect. 4.

Preliminaries
Some notations are given first. For 1 ≤ s ≤ +∞, L s (Ω) denotes a Lebesgue space with the norm given by | · | s . Let D 1,2 (R 3 ) be the completion of C ∞ 0 (R 3 ) endowed with the norm The space H 1 (R 3 ) is endowed with the following standard product and norm, respectively: The best Sobolev constant S is given by For any ρ > 0 and x ∈ R 3 , B ρ (x) denotes the ball of radius ρ centered at x. As pointed out in [4], the existence of solutions are not related to the signs of ω, so one can assume that ω > 0. Similar to [12], we now first give the variational structure of system (1.1). Let be a Hilbert space, whose inner product and norm are given by respectively. For λ ≥ 1, the inner product and norm are defined as It is obvious u ≤ u λ for λ ≥ 1. Let H λ = (H, · λ ). By (A1)-(A2) and the Poincaré inequality, the embedding H λ → H 1 (R 3 ) is continuous. Thus, for s ∈ [2,6], there exists γ s > 0 which is independent of λ such that the corresponding bilinear form of A λ is defined as As in [23], for fixed λ > 0, consider the following eigenvalue problem: Hence, following [24], the following proposition is obtained in [12]. Then System (1.1) has a variational formulation. Actually, the corresponding functional ϕ λ : The pair (u, φ) ∈ H λ × D 1,2 (R 3 ) is a solution of system (1.1) if and only if it is a critical point of ϕ λ . By borrowing the reduction method used in [25], we can study ϕ λ (u, φ) with only one variable u. The following technical result comes from [12].

Proposition 2.3 ([12])
Let K(x) satisfy the condition (K). Then, for any u ∈ H λ , there exists a unique φ = φ u ∈ D 1,2 (R 3 ) which satisfies Remark 2.4 It is pointed out in [12] that the condition (K) can be replaced by Multiplying both sides of the equationφ u + K(x)φ u u 2 = -ωK(x)u 2 by φ u and integrating by parts, we get , the Gateaux derivative of Ψ λ (u) is given by The properties of the functional G is given by [12], the derivative G possesses the Brezis-Lieb-splitting (written for BL-splitting) property, which is similar to the Brezis-Lieb lemma [26].
Next, the compactness conditions for the functional Ψ λ is considered. It is well known that a C 1 functional I satisfying Palais-Smale ((PS) for short) condition at level c if for any sequence {u n } ⊂ H such that I(u n ) → c and I (u n ) → 0, there exists a convergent subsequence of H, which is called a (PS) c sequence. Since K ∈ L ∞ (R 3 ) and K ∈ L 3 (R 3 ) are similar, in the following, we just consider the case K ∈ L ∞ (R 3 ). Proof Let {u n } ⊂ H λ be a (PS) c sequence of Ψ λ . Arguing indirectly, suppose u n λ → ∞ such that after passing to a subsequence. Denote w n := u n / u n λ . Then w n λ = 1, w n w 0 in H λ and w n (x) → w 0 (x) for a.e. x ∈ R 3 . If then 0 ≥ μ 2 -1, which contradicts μ > 2. Case (2). μ ≥ 4. In this case, by (2.4), (2.5) and (2.6), we have then 0 ≥ μ 2 -1, which contradicts μ ≥ 4.
Since w n = u nu, so u n → u in H λ .
However, since the appearance of the nonlinear term h in Ψ λ (u) and A is sign-changing, we cannot deduce that Ψ λ (u) ≥ 0 from So there are two situations to consider: (i) Ψ λ (u) < 0; (ii) Ψ λ (u) ≥ 0.
Proof of Theorem 1.3 The first solution can be proved in the same way as shown in Theorem 1.2. The second solution follows from [12, Lemma 3.2, Lemma 3.5], Lemma 2.6, Lemma 2.7 and Lemma 3.1.

Conclusion
In this paper, we first obtained a Palais-Smale sequence by using super-quadratic condition. Then we establish the parameter which depends on compactness conditions to recover the compactness. Finally, the existence of nontrivial solutions is proved by the linking theorem and Ekeland's variational principle. Obviously, the super-quadratic condition has been successfully applied to find the solutions of the nonhomogeneous Klein-Gordon-Maxwell system with sign-changing potential, we hope that these results can be widely used in fractional systems as discussed in [29] and [30].