Multiple positive solutions of a quasilinear Schrödinger-Poisson system with concave and convex nonlinearities

Abstract

In this paper, we consider the quasilinear Schrödinger-Poisson system with concave and convex nonlinearities \begin{align*} \begin{cases} -Δ_{p} u+λV(x)|u|^{p-2}u + μϕ|u|^{p-2}u= a(x)|u|^{m-2}u + b(x)|u|^{q-2}u & \ \ \ \mathrm{in}\ \mathbb{R}^{3}, -Δϕ=|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 ${\Delta }_{p}u=div(|\nabla u{|}^{p-2}\nabla u)$. 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.