Existence and asymptotic behaviour of standing waves for quasilinear Schrödinger–Poisson systems in $R^3$

Abstract

In this work we are concerned with existence and asymptotic behaviour of standing wave solutions in the whole space ${\mathbb{R}}^3$ for the quasilinear Schrödinger–Poisson system

when the nonlinearity coefficient $\varepsilon >0$ goes to zero. Under appropriate, almost optimal, assumptions on the potential $\tilde{V}$ and the density $n^{\ast }$ we establish existence of a ground state $(u_{\varepsilon },V_{\varepsilon })$ of the above system, for all ɛ sufficiently small, and show that $(u_{\varepsilon },V_{\varepsilon })$ converges to $(u_0,V_0)$, the ground state solution of the corresponding system for $\varepsilon =0$.