Generalized Schrödinger-Poisson type systems

In this paper we study the boundary value problem \[ \left\{ \begin{array}{ll} -\Delta u+ \eps q\Phi f(u)=\eta|u|^{p-1}u & \text{in } \Omega, \\ - \Delta \Phi=2 qF(u)& \text{in } \Omega, \\ u=\Phi=0 & \text{on }\partial \Omega, \end{array} \right.\] where $\Omega \subset \mathbb{R}^3$ is a smooth bounded domain, $1 < p < 5$, $\eps ,\eta= \pm 1$, $q>0$, $f:\R\to\R$ is a continuous function and $F$ is the primitive of $f$ such that $F(0)=0.$ We provide existence and multiplicity results assuming on $f$ a subcritical growth condition. The critical case is also considered and existence and nonexistence results are proved.