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.
Categories: math.AP