Infinitely many solutions and concentration of ground state solutions for the Klein-Gordon-Maxwell system

Abstract In this paper, we consider a model which describes solitary waves of the nonlinear Klein-Gordon equation interacting with the electromagnetic field, namely, the following Klein-Gordon-Maxwell system { − Δ u + ( m 2 − ω 2 ) u − ( 2 ω + ϕ ) ϕ u = f ( u ) , in R 3 , − Δ ϕ = − ( ω + ϕ ) u 2 , in R 3 , where 0 ω m are constants and the nonlinearity f satisfies the superlinear condition. Here m and ω denote the mass and the phase respectively, while u and ϕ are unknowns. By using variational methods and some technique related to Pohozǎev identity, we construct bounded Palais-Smale sequences to obtain the existence of infinitely many high energy radial solutions. Moreover, we show that the ground state solutions of the Klein-Gordon-Maxwell system tend to the ground state solutions of the classical Schrodinger equation as ω → 0 .