The quasilinear Schrödinger--Poisson system

Abstract

This paper deals with the $(p,q)$--Schrödinger--Poisson system \begin{eqnarray*} \left \{\begin{array}{ll} \displaystyle -Δ_p u+|u|^{p-2}u+λϕ|u|^{s-2}u=|u|^{r-2}u,&\mathrm{in} \ \mathbb{R}^3,\\ \displaystyle -Δ_q ϕ= |u|^s, &\mathrm{in}\ \mathbb{R}^3,\\ \end{array} \right. \end{eqnarray*} where $1<p<3$, $\max \left\{1,\frac{3p}{5p-3}\right\}<q<3$, $p<r<p^{\ast }:=\frac{3p}{3-p}$, $\max \left\{1,\frac{(q^{\ast }-1)p}{q^{\ast }}\right\}<s<\frac{(q^{\ast }-1)p^{\ast }}{q^{\ast }}$, ${\Delta }_{i}u=div(|\nabla u{|}^{i-2}\nabla u)(i=p,q)$ and $\lambda >0$ is a parameter. This quasilinear system is new and has never been considered in the literature. The uniqueness of solutions of the quasilinear Poisson equation is obtained via the Minty--Browder theorem. The variational framework of the quasilinear system is built and the nontrivial solutions of the system are obtained via the mountain pass theorem.