The quasilinear Schr\"odinger--Poisson system

This paper deals with the $(p,q)$--Schr\"odinger--Poisson system \begin{eqnarray*} \left \{\begin{array}{ll} \displaystyle -\Delta_p u+|u|^{p-2}u+\lambda\phi |u|^{s-2}u=|u|^{r-2}u,&\mathrm{in} \ \mathbb{R}^3,\\ \displaystyle -\Delta_q \phi = |u|^s, &\mathrm{in}\ \mathbb{R}^3,\\ \end{array} \right. \end{eqnarray*} where $1<p<3$, $\max \left\{1,\frac{3p}{5p-3}\right\}<q<3$, $p<r<p^*:=\frac{3p}{3-p}$, $\max\left\{1,\frac{(q^*-1)p}{q^*}\right\}<s<\frac{(q^*-1)p^*}{q^*}$, $\Delta_i u=\hbox{div}(|\nabla u|^{i-2}\nabla u)\ (i=p,q)$ 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.