Multiple positive solutions of a quasilinear Schr\"{o}dinger-Poisson system with concave and convex nonlinearities

Abstract

In this paper, we consider the quasilinear Schr\"{o}dinger-Poisson system with concave and convex nonlinearities \begin{align*} \begin{cases} -\Delta_{p} u+\lambda V(x)|u|^{p-2}u + \mu \phi |u|^{p-2}u= a(x)|u|^{m-2}u + b(x)|u|^{q-2}u & \ \ \ \mathrm{in}\ \mathbb{R}^{3}, -\Delta \phi=|u|^{p} &\ \ \ \mathrm{in}\ \mathbb{R}^{3}, \end{cases} \end{align*} where $\lambda >0, ~\mu >0$, $\frac{3}{2}<p<3$, $1< q<p < m < 2p$ and $\Delta_{p} u= \hbox{div}(|\nabla u|^{p-2}\nabla u)$. We assume that $V(x) \in C(\mathbb{R}^{3}, \mathbb{R})$ is a steep potential well, while $a(x)$ and $b(x)$ are allowed to be sign-changing and satisfy some suitable assumptions in $\mathbb{R}^3$. By using the Ekeland's variational principle and combining the constraint approach, we prove that the system admits two positive solutions.