Existence of a solution to a nonlinear system of Schrödinger--Poisson type with sublinear eigenfunction term

In this paper, we prove the existence of a nontrivial solution to a system, whose model is \begin{equation*} \begin{cases} - \Delta u + r \psi \ u^{r-1} = u^{p-1} \qquad & \mbox{ in } \Omega\,,\\ - \Delta \psi = u^{r} \qquad & \mbox{ in } \Omega\,,\\ u , \, \psi > 0 & \mbox{ in } \Omega\,, \\ u = \psi =0 & \mbox{ on } \partial\Omega\,, \end{cases} \end{equation*} where $\Omega$ is bounded domain in $\mathbb R^N$, $N\geq 2$, with $p\in(1,2)$ and $r\geq1$. Such a solution turns out to be a saddle point of a suitable functional.