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: equation(P){−Δu+[m2−(eϕ+ω)2]u=f(u),in R3,Δϕ=e(eϕ+ω)u2,in R3, where m,e,ω>0m,e,ω>0. Benci and Fortunato (2002) [3] and D’Aprile and Mugnai (2004) [6], showed that, for any u∈H1(R3)u∈H1(R3), the second equation of problem (P) has a unique solution ϕu∈D1,2(R3)ϕu∈D1,2(R3), the map Λ:u∈H1(R3)↦ϕu∈D1,2(R3) is continuously differentiable, and ϕu∈[−ω/e,0]ϕu∈[−ω/e,0]. Furthermore, we prove that max{−ωe−ϕu,ϕu}≤ψu≤0, where ψu=Λ′(u)[u]/2. Then, we consider the ground-state solution of problem (P) with f(u)=|u|p−2u,2