Existence Result for Fractional Klein-Gordon-Maxwell System with Quasicritical Potential Vanishing at Infinity

The following fractional Klein-Gordon-Maxwell system is studied 
 
 
(-Δ)p stands for the fractional Laplacian, ω > 0 is a constant, V is vanishing potential and K is a smooth function. Under some suitable conditions on K and f, we obtain a Palais-Smale sequence by using a weaker Ambrosetti-Rabinowitz condition and prove the ground state solution for this system by employing variational methods. In particular, this kind of problem is a vast range of applications and challenges.


Introduction
In this paper, the following fractional Klein-Gordon-Maxwell system is considered This kind of problem can apply to various fields. For example, Li et al. [3] studied a class of fractional Schrödinger equation with potential vanishing at infinity by using variational methods and obtained a positive solution for this equation.
, system (1.1) reduces to a Klein-Gordon-Maxwell equation, which was first studied by Benci and Fortunato [7] as a model describing a nonlinear Klein-Gordon equation interacting with an electromagnetic field with 4 6 q < < . For more details on the physical aspects of this problem, we refer the readers to see [8] and references therein. , , , , they obtained some results which complete the results obtained in [7].
In recent years, under various hypotheses on the potential ( ) V x and the nonlinearity ( ) f u , the existence of positive, multiple, ground state solutions for Klein-Gordon-Maxwell systems or similar systems, has been widely studied in the literature. For example, Azzollini and Pomponio [10] first proved the existence of a ground state solution for system (1.2) when the nonlinearity is more general. He [11] first considered a Klein-Gordon-Maxwell system with non-constant potential. Li and Tang [12] improved the result of [11]. A nonlinear Klein-Gordon-Maxwell system with sign-changing potential was first considered by Ding and Li in [13]. They obtained infinitely many solutions by symmetric mountain pass theorem. Otherwise, there are many works about the nonhomogeneous Klein-Gordon-Maxwell system. Wang [14] proved that a nonhomogeneous Klein-Gordon-Maxwell system had two solutions. In [15], Gan et al.
. Inspired mainly by the aforementioned results, we find a ground state solution for (1.1) with potential vanishing at infinity. To show our result, we make the following assumptions first: To the best of our knowledge, Ambrosetti-Rabinowitz condition (AR condition for short) plays an important role in proving the boundedness of Palais-Smale sequence (PS sequence for short). In recent years, there are many papers devoted to replacing (AR) condition with weaker condition. It is easy to see that (H4) is weaker than (H4'). In this paper, we obtain a (PS) sequence by using the weaker (AR) condition. Besides, it seems that there is only one work about the Klein-Gordon-Maxwell system involving fractional Laplacian. In this paper, the main difficulty is lack of compactness of Sobolev embedding in whole space because of the nonlocal term φ and the fractional operator. To overcome this problem, we use the reduction method introduced by Caffarelli and Silvestre [17] and recover the compactness by the interaction of the behaviour of the potential and nonlinearity. This paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of main result.

Preliminaries
In this section, by the local reduction derived from Caffarelli and Silvestre [17], we first reformulate the nonlocal fractional system (1.1) into a local system, that is ,0 : u x w x w = =  , and , , where  denotes the Fourier transform, that is where j denotes the imaginary unit. When ϕ is smooth enough, the ( ) which is of 1 C by (H1)-(H3).
 in E, then up to a subsequence, for any z E ∈ , after passing to a subsequence. Denote :