Ground-state solutions for the electrostatic nonlinear Klein–Gordon–Maxwell system

In this paper, we study the nonlinear Klein–Gordon equation coupled with the Maxwell equation in the electrostatic case: (P) { − Δ u + [ m 2 − ( e ϕ + ω ) 2 ] u = f ( u ) , in R 3 , Δ ϕ = e ( e ϕ + ω ) u 2 , in R 3 , where m , e , ω > 0 . Benci and Fortunato (2002) [3] and D’Aprile and Mugnai (2004) [6], showed that, for any u ∈ H 1 ( R 3 ) , the second equation of problem (P) has a unique solution ϕ u ∈ D 1 , 2 ( R 3 ) , the map Λ : u ∈ H 1 ( R 3 ) ↦ ϕ u ∈ D 1 , 2 ( R 3 ) is continuously differentiable, and ϕ u ∈ [ − ω / e , 0 ] . Furthermore, we prove that max { − ω e − ϕ u , ϕ u } ≤ ψ u ≤ 0 , where ψ u = Λ ′ ( u ) [ u ] / 2 . Then, we consider the ground-state solution of problem (P) with f ( u ) = | u | p − 2 u , 2 < p < 6 .