Multiple positive solutions for a Schrödinger-Poisson-Slater system

Abstract

In this paper we investigate the existence of positive solutions to the following Schrödinger-Poisson-Slater system [c]{ll} - Δu+ u + λϕu=|u|^{p-2}u & \text{in} Ω-Δϕ= u^{2} & \text{in} Ωu=ϕ=0 & \text{on} \partialΩ. where $\Omega$ 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 $\Omega$.

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