Multiple positive solutions for a Schr\"odinger-Poisson-Slater system

In this paper we investigate the existence of positive solutions to the following Schr"odinger-Poisson-Slater system [c]{ll} - \Delta u+ u + \lambda\phi u=|u|^{p-2}u & $\text{in}$ \Omega -\Delta\phi= u^{2} & $\text{in}$ \Omega u=\phi=0 & $\text{on}$ \partial\Omega. where $\mathrm{\Omega }$ is a bounded domain in ${\mathbf{R}}^3,\lambda$ is a fixed positive parameter and $p<2^{\ast }=\frac{2N}{N-2}$. We prove that if $p$ is “near” the critical Sobolev exponent $2^{\ast }$, then the number of positive solutions is greater then the Ljusternik-Schnirelmann category of $\mathrm{\Omega }$.