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

Canlin Gan,Ting Xiao,Qiongfen Zhang

doi:10.4236/jamp.2020.87101

Abstract

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.

Highlights

In this paper, the following fractional Klein-Gordon-Maxwell system is considered (−∆)p u +V ( x)u − (2ω + φ )φu = K ( x) f (u), in 3, )u2, in 3, (1.1)where p ∈ (3 4,1), (−∆)p denotes the fractional Laplacian operator, V is zero mass potential and K is a smooth function
The fractional Schrödinger equation was first proposed by Laskin [1] [2] as a result of expanding the Feynman
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

Summary

Introduction

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. Obtained two solutions for a type of nonhomogeneous Klein-Gordon-Maxwell system with sign-changing potential Another example is [16], Miyagaki et al investigated system (1.1) with fractional Laplacian and f satisfied the following type of Ambrosetti-Rabinowitz condition:. We obtain a (PS) sequence by using the weaker (AR) condition It seems that there is only one work about the Klein-Gordon-Maxwell system involving fractional Laplacian.

Preliminaries

Proof of Main Result

Conclusion