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 − Δ p u − l (x) ϕ | u | p − 2 u = | u | p * − 2 u + μ h (x) | u | q − 2 u in R 3 and −Δϕ = l(x)|u|p in R 3 , where μ > 0, p * = 3 p 3 − 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 3 2 < p < 3. [ABSTRACT FROM AUTHOR]