Semiclassical solutions for the nonlinear Schrödinger–Maxwell equations

In this paper, by using variational methods and critical point theory, we study the existence of semiclassical solutions for the following nonlinear Schrödinger–Maxwell equations { − ε 2 Δ u + V ( x ) u + ϕ u = f ( x , u ) , in R 3 , − Δ ϕ = 4 π u 2 , in R 3 , where ε > 0 and V ( x ) ⩾ 0 for all x ∈ R 3 . Under some weaker assumptions on V and f, we prove that the system has at least one nontrivial solution for sufficient small ε > 0 .