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

Abstract

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 $\Omega$.