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 { − Δ u + u + λ ϕ u = | u | p − 2 u in Ω , − Δ ϕ = u 2 in Ω , u = ϕ = 0 on ∂ Ω , where Ω is a bounded domain in R 3 , λ is a fixed positive parameter and p < 2 ∗ = 2 N N − 2 . We prove that if p is “near” the critical Sobolev exponent 2 ∗ , then the number of positive solutions is greater then the Lusternik–Schnirelmann category of Ω.