Electronic Journal of Qualitative Theory of Differential Equations

In this paper we study the following nonlinear Klein–Gordon–Maxwell system { −∆u + [m0 − (ω + φ)2]u = f (u) in R3, ∆φ = (ω + φ)u in R3, where 0 < ω < m0. Based on an abstract critical point theorem established by Jeanjean, the existence of positive ground state solutions is proved, when the nonlinear term f (u) exhibits linear near zero and a general critical growth near infinity. Compared with other recent literature, some different arguments have been introduced and some results are extended.


Introduction
This article is concerned with the following Klein-Gordon-Maxwell equations where 0 < ω < m 0 . We assume that the followings hold for f : ( f 4 ) there exist D > 0 and q ∈ (2, 6) such that f (s) + ms ≥ Ks 5 + Ds q−1 for all s > 0; ( f 5 ) there exists constant γ > 2 such that f (s)s − γF(s) ≥ 0 for all s ∈ R, where F(s) = s 0 f (t)dt. This system is well known as a model describing the interaction between the nonlinear Klein-Gordon field and the electrostatic field. The presence of nonlinear term f (u) simulates the interaction between many particles or external nonlinear perturbations.
In recent years, there is large quality works devoted to the system (KGME), and we would like to recall some of them. In a remarkable work, V. Benci and D. Fortunato [4] are the first to study the following system using the variational method, the authors proved the existence of infinitely many radially symmetric solutions when m 0 > ω > 0 and 4 < q < 6. In [15,16], D'Aprile and Mugnai considered the case 2 < p ≤ 4 and established some non-existence results for p > 6. Afterwards, there are also more literatures focusing on the existence and multiplicity of solutions for the problem (KGME). See [12,13,19] and the references therein.
Soon afterwards, the authors of [9] studied the following critical Klein-Gordon-Maxwell system with external potential: Provided f (u) satisfying assumptions: They obtained a nontrivial solution for (1.3). For more related results, we refer the readers to [3,24].
The existence of ground state solutions, that is, couples (u, φ) which solve (KGME) and minimize the action functional associated to (KGME) among all possible nontrivial solutions, has been investigated by many authors. Inspired by the approach of Benci and Fortunato, Azzollini and Pomponio [10] proved that (1.1) admits a ground state solution provided one of the following assumptions: (i) 3 ≤ p < 5 and m 0 > ω; Soon afterwards, Carrião et al. [22] dealt with the critical Klein-Gordon-Maxwell system (1.2) with potentials. Combining the minimization of the corresponding Euler-Lagrange functional on the Nehari manifold, they proved the existence of positive ground state solutions for system (1.2). Very recently, Moura, Miyagaki et al. [14] considered quasicritical Klein-Gordon-Maxwell systems with potential, and obtained positive ground state solutions. For other related results about Klein-Gordon-Maxwell systems the authors maybe see [7,17,25].
Here we also mention that the papers [2,6], Berestycki and Lions studied the following elliptic equation Under the following conditions on f (u): Berestycki and Lions [6] proved the existence of a positive least energy solution when N ≥ 3 and Berestycki et al. [2] investigated the existence of infinitely many bound state solutions when N = 2. Under the above assumptions, Azzollini, d'Avenia and Pomponio [1] obtained the existence of at least a radial positive solution to a class of Schrödinger-Poisson problems, and Azzollini [28] proved the existence of ground state solutions for Kirchhoff-type problems, and soon after Zhang and Zou [27] investigated the existence of ground state solutions of the problem (1.4) with the critical growth assumption on f (u).
Under the assumptions ( f 1 )-( f 5 ), Zhang [21] studied a class of Schrödinger-Poisson problems and established the existence of ground state solutions for q ∈ (2, 4] with D large enough, or q ∈ (4, 6), where m = 0; Liu [20] considered a Kirchhoff-type problem and obtained the existence of ground state solutions without ( f 5 ).
Motivated by the above mentioned works, in particular by [9,20,21,27], the main purpose of this paper is to consider the existence of positive least energy solutions of (KGME) with a general nonlinearity in the critical growth. To our best knowledge, under the assumptions ( f 1 )-( f 5 ), there is no work on the the existence of positive ground state solutions for problem (KGME). Precisely, we have the following results.
Assume that either q ∈ (2, 4] with D sufficiently large, or q ∈ (4, 6), then the problem (KGME) possesses a positive radial solution if one of the following conditions is satisfied: Assume that either q ∈ (2, 4] with D sufficiently large, or q ∈ (4, 6), then the problem (KGME) possesses a positive ground state solution provided one of the following conditions holds: Theorem 1.3. If we replace the condition ( f 5 ) by the following condition: Then the conclusions of Theorems 1.1 and 1.2 remain true. [20,21]. Since the problem in [20] is different from ours, the methods used in [20] do not work here. The similar hypotheses on f (u) as above ( f 1 )-( f 5 ) are introduced in [21], where the authors used a cut-off functional to obtain bounded (PS) sequences. However, our device is different from the main arguments of [21]. Moreover, the results of [21] hold under γ > 3, and in our case, γ > 2.

Remark 1.5.
The condition ( f 4 ) plays a crucial role to ensure the existence of ground state solution to the problem (KGME). And the condition ( f 5 ) is a technical condition to overcome the difficulty caused by the critical exponential growth case.
In our paper, due to the presence of a nonlocal term φ and the effect of the nonlinearity in the critical growth, there exist several difficulties to solve. In the first place, the lack of the following Ambrosetti-Rabinowitz growth hypothesis on f : brings a obstacle in proving the boundedness of (PS) sequence. To overcome this difficulty, we will use approaches developed by Jeanjean [23] to obtain the boundedness. In the next place, since we deal with the critical case, the Sobolev embedding H 1 (R 3 ) → L 6 (R 3 ) is not compact, and the functional I does not satisfy (PS) c condition at every energy level c. To avoid the difficulty, we try to pull the energy level down below some critical level c * 1 (Section 3). In the end, we apply the Strauss' compactness result [5] to obtain the convergence of (PS) c sequence.
An outline of the paper is as follows. In Section 2, we give some preliminary lemmas. Section 3 is devoted to the existence of the mountain pass solution and positive ground state solution. Throughout the paper we denote by C the various positive constants. Let 1 3 denotes the best Sobolev constant.

Preliminaries
In this section we give notations and prove some preliminary lemmas. Let us define an equivalent norm on H 1 (R 3 ), that is For any 1 ≤ s < ∞, we denote that L s (R 3 ) is the usual Lebesgue space endowed with the norm ∥u∥ s L s = R 3 |u| s dx. Then we have that, According to the variational nature of (KGME), we define its the energy functional as follows: and that the weak solutions of (KGME) is critical points of the functional Φ. Obviously, the functional Φ is the strongly indefiniteness, which means that it is unbounded both from below and from above on infinite-dimensional subspaces. In order to avoid this indefiniteness, we apply the reduction method developed by Benci and Fortunato [8]. For deducing our results, we introduce the following results whose idea of proof comes from [15,16].
Multiplying (2.2) by φ u and integrating by parts, we obtain By the definition of Φ and (2.3) the functional I(u) = Φ(u, φ) may be rewritten as the following form In view of Lemmas 2.1 and 2.2, the conditions ( f 1 )-( f 3 ) imply I(u) ∈ C 1 and its Gateaux derivative is for all u, v ∈ H. Then (u, φ) is a weak solution of (KGME) if and only if φ = φ u and u is a critical point of I on H.
For simplicity, in this paper we may assume that K = 1. Set g(t) = f (t) + mt, so the functional I is reduced as where G(s) = s 0 g(t)dt. In the following we give the abstract result established by Jeanjean [23].
If for every λ ∈ h the set Γ λ is nonempty and Then for every almost λ ∈ h there is a sequence {u n } ⊂ X such that and so the family of functionals we study is and for every u, v ∈ H, We shall use the following Pohožaev type identity. Its proof can be done as in [16].
If λ = 1, the above Pohožaev equality turns to be the following Next we shall cite a variant of the Strauss compactness result [5], which plays a fundamental tool in our arguments: Lemma 2.5. Let P and Q : R → R be two continuous functions satisfying Let {v n } n , v and ψ be measurable functions from R N to R, with z bounded, such that a.e. in R N , as n → +∞.
Then for any bounded Borel set B one has ∥(P

Proof of main results
In this section we will look for a positive ground state solutions of (KGME). First, we will prove the existence of a mountain pass solution. Now, we give several lemmas which imply that I λ satisfies the conditions of Lemma 2.3.
It follows from Lemma 3.1 that the conclusions of Lemma 2.3 hold.
Note that I(u n ) = I λ n (u n )−(λ n −1) R 3 F(u n )dx and I ′ (u n ) = I ′ λ n (u n )−(λ n −1) R 3 f (u n )u n dx. By using the fact that the map λ → c λ is left-continuous (see [23]), λ n → 1, the boundedness of {u n }, we can show that where u + = max{u, 0}. Repeating all the calculations above word by word, there is nonzero function u 0 solving the equation −∆u + (m 2 0 − ω 2 )u + mu − ωφ u u = (g(u) − u 5 ) + (u + ) 5 . (3.36) Using u − = max{−u 0 , 0} as a text function and integrating (3.36) by parts, we obtain (3.37) We deduce from ( f 1 ) and ( f 4 ) that g(t) − t 5 is an odd function and g(t) − t 5 > 0 for t > 0. So from (3.37) one has From Lemma 2.1, we obtain that u − 0 = 0 and u 0 ≥ 0. Then u 0 is a nonnegative solution of the problem (KGME). Deducing from Harnack's inequality (see [26]), we can obtain that u 0 > 0 for all x ∈ R 3 , and u 0 is a positive critical point of the functional I(u). Then by Lemma 2.1, we have φ = φ u 0 . From (2.1), (2.3) and (2.4) that (u 0 , φ u 0 ) is a positive solution of (KGME). The proof is complete. In what follows, we prove the existence of a positive ground state solution for the problem (KGME).