Multiple positive solutions for a Schrödinger–Poisson–Slater system

In this paper we investigate the existence of positive solutions to the following Schrödinger–Poisson–Slater system{−Δu+u+λϕu=|u|p−2uin Ω,−Δϕ=u2in Ω,u=ϕ=0on ∂Ω, where Ω   is a bounded domain in R3R3, λ   is a fixed positive parameter and p<2∗=2NN−2. We prove that if p   is “near” the critical Sobolev exponent 2∗2∗, then the number of positive solutions is greater then the Lusternik–Schnirelmann category of Ω.