This paper is concerned with the quasilinear Schrödinger–Poisson system Δpul(x)ϕ|u|p2u=|u|p*2u+μh(x)|u|q2u in R3 and −Δϕ = l(x)|u|p in R3, where μ > 0, p*=3p3p and Δpu = div(|∇u|p−2u). 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 pq < p* and 32<p<3.

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