Existence and asymptotic behavior of nontrivial solution for Klein–Gordon–Maxwell system with steep potential well

In this paper, we consider the following nonlinear Klein–Gordon–Maxwell system with a steep potential well { − Δ u + ( λ a ( x ) + 1 ) u − μ ( 2 ω + ϕ ) ϕ u = f ( x , u ) , in R 3 , Δ ϕ = μ ( ω + ϕ ) u 2 , in R 3 , where ω > 0 is a constant, μ and λ are positive parameters, f ∈ C ( R 3 × R , R ) and the nonlinearity f satisfies the Ambrosetti–Rabinowitz condition. We use parameter-dependent compactness lemma to prove the existence of nontrivial solution for μ small and λ large enough, then explore the asymptotic behavior as μ → 0 and λ → ∞ . Moreover, we also use truncation technique to study the existence and asymptotic behavior of positive solution of Klein–Gordon–Maxwell system when f ( u ) := | u | q − 2 u where 2 < q < 4 .