Existence and Asymptotic Behaviour of Solutions for a Quasilinear Schrödinger-Poisson System in ${\mathbb {R}}^3$

Abstract

In this paper, we study the existence and asymptotic behaviour of solutions for the following quasilinear Schrödinger-Poisson system in ${\mathbb {R}}^3$

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+V(x)u+\lambda \phi u=f(x,u),&{}x\in {{\mathbb {R}}^3},\\ -\Delta \phi -\varepsilon ^4\Delta _4\phi =\lambda u^2,&{}x\in {{\mathbb {R}}^3}, \end{array}\right. \end{aligned}$$

where $\lambda$ and $\varepsilon$ are positive parameters, $\Delta _4=\hbox {div}(|\nabla u|^2\nabla u)$, V is a continuous and coercive potential function with positive infimum, f is a Carathéodory function defined on ${\mathbb {R}}^3\times {\mathbb {R}}$ satisfying the classic Ambrosetti-Rabinowitz condition. First, a nontrivial solution is obtained for $\lambda$ small enough and $\varepsilon$ fixed by variational methods and truncation technique. Later, the asymptotic behaviour of these solutions is studied whenever $\varepsilon$ and $\lambda$ tend to zero respectively. We prove that they converge to a nontrivial solution of a classic Schrödinger-Poisson system and a class of Schrödinger equation associated respectively.