  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 $\Omega$ is a bounded domain
in $\mathbf{R}^{3},\lambda$ is a fixed positive parameter and
$p<2^{*}=\frac{2N}{N-2}$. We prove that if $p$ is "near" the critical Sobolev
exponent $2^*$, then the number of positive solutions is greater then the
Ljusternik-Schnirelmann category of $\Omega$.
