Multiple positive solutions of the quasilinear Schrödinger–Poisson system with critical exponent in D1,p(R3)

This paper is concerned with the quasilinear Schrödinger–Poisson system −Δpu−l(x)ϕ|u|p−2u=|u|p*−2u+μh(x)|u|q−2u in R3 and −Δϕ = l(x)|u|p in R3, where μ > 0, p*=3p3−p and Δpu = div(|∇u|p−2∇u). By using the Ekeland’s variational principle and the mountain pass theorem, we prove that the system admits two positive solutions for 1 ⩽ q < p and 1 < p < 3, and the system admits one positive solution for p ⩽ q < p* and 32<p<3.