On ground state solutions for the Schrödinger–Poisson equations with critical growth

In this paper, we study the existence of ground state solutions for the following Schrödinger–Poisson equation { − Δ u + V ( x ) u + λ ϕ u = μ | u | q − 1 u + | u | 4 u , in R 3 , − Δ ϕ = u 2 , in R 3 , where μ is a positive parameter. Under some certain assumptions on V, we prove that for every λ > 0 and q ∈ ( 2 , 5 ) , such a class of Schrödinger–Poisson equation with critical growth has at least a positive ground state solution via variational methods. Some recent results from the literature are extended.