Existence and asymptotic behaviour of solutions for a quasi-linear Schrödinger–Poisson system with a critical nonlinearity

In this paper we consider the following quasilinear Schrödinger–Poisson system {-Δu+u+ϕu=λf(x,u)+|u|4uinR3-Δϕ-ε4Δ4ϕ=u2inR3,depending on the two parameters λ, ε> 0. We first prove that, for λ larger than a certain λ∗> 0 , there exists a solution for every ε> 0. Later, we study the asymptotic behaviour of these solutions whenever ε tends to zero, and we prove that they converge to the solution of the Schrödinger–Poisson system associated.