Ground state solutions for Schrödinger–Poisson system with critical growth and nonperiodic potential

This paper is concerned with the existence of ground state solutions for the Schrödinger–Poisson system −Δu + V(x)u + ϕu = |u|4u + λ|u|p−2u in R3 and −Δϕ = u2 in R3, where λ > 0 and p ∈ [4, 6). Here, V(x)∈C(R3,R), V(x) = V1(x) for x1 > 0, and V(x) = V2(x) for x1 < 0, where V1, V2 are periodic functions in each coordinate direction. In this paper, we give a splitting lemma corresponding to the nonperiodic potential and, then, prove the existence of ground state solutions for any λ > 0 when p ∈ (4, 6). Moreover, when p = 4, the above system possesses a ground state solution for λ > 0 sufficiently large. It is worth underlining that the technique employed in this paper is also valid for the Sobolev subcritical problem studied by Cheng and Wang [Discrete Contin. Dyn. Syst., Ser. B 27, 6295 (2022)].