Existence and concentration behavior of solutions for the logarithmic Schrödinger–Poisson system with steep potential

In this paper, we study the following logarithmic Schrödinger–Poisson system: {-Δu+λV(x)u+q(x)ϕu=ulogu2,inR3,-Δϕ=q(x)u2,inR3,where λ> 0 , V(x) ∈ C(R, R) and q(x) ≥ 0 for all x∈ R. Under the suitable conditions on potential V(x) and q(x), by proceeding a new penalization scheme to the nonlocal term ϕ, combining with a new version of Minimax method, we prove the existence of positive solution u∈ H(R) of the above system for λ> 0 large enough. Moreover, we also investigate the concentration behavior of { u} as λ→ + ∞.