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

In this paper we consider the following quasilinear Schrodinger-Poisson system {-Delta u + u +phi u = lambda f(x, u) + vertical bar u vertical bar(4)u on R-3 {-Delta phi + epsilon(4) Delta(4)phi = u(2) in R-3,R- depending on the two parameters lambda epsilon > 0. We first prove that, for. larger than a certain lambda* > 0, there exists a solution for every epsilon > 0. Later, we study the asymptotic behaviour of these solutions whenever epsilon tends to zero, and we prove that they converge to the solution of the Schrodinger-Poisson system associated.