In this paper, we study the existence and multiplicity solutions for the following Klein–Gordon–Maxwell system { − Δ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = f ( x , u ) , x ∈ R 3 , Δ ϕ = ( ω + ϕ ) u 2 , x ∈ R 3 , where ω > 0 is a constant and the nonlinearity f ( x , u ) is either asymptotically linear in u at infinity or the primitive of f ( x , u ) is of 4-superlinear growth in u at infinity. Under some suitable assumptions, the existence and multiplicity of solutions are proved by using the Mountain Pass theorem and the fountain theorem, respectively.

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