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 Schr\"odinger-Poisson system $$ \left\{ \begin{array}[c]{ll} - \Delta u +u+\phi u = \lambda f(x,u)+|u|^{2^{*}-2}u &\ \mbox{in } \mathbb{R}^{3} \\ -\Delta \phi -\varepsilon^{4} \Delta_4 \phi = u^{2} & \ \mbox{in } \mathbb{R}^{3}, \end{array} \right. $$ depending on the two parameters $\lambda,\varepsilon>0$. We first prove that, for $\lambda$ larger then a certain $\lambda^{*}>0$, there exists a solution for every $\varepsilon>0$. Later, we study the asymptotic behaviour of these solutions whenever $\varepsilon$ tends to zero, and we prove that they converge to the solution of the Schr\"odinger-Poisson system associated.