Existence and asymptotic behaviour of solutions for a quasi-linear schrodinger-poisson system under a critical nonlinearity

Abstract

In this paper we consider the following quasilinear Schrödinger-Poisson system $$ \left\{ \begin{array}[c]{ll} - Δu +u+ϕu = λf(x,u)+|u|^{2^{*}-2}u &\ \mbox{in } \mathbb{R}^{3} \\ -Δϕ-\varepsilon^{4} Δ_4 ϕ= u^{2} & \ \mbox{in } \mathbb{R}^{3}, \end{array} \right. $$ depending on the two parameters $\lambda ,\epsilon >0$. We first prove that, for $\lambda$ larger then a certain ${\lambda }^{\ast }>0$, 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 Schrödinger-Poisson system associated.