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 R 3 and −Δϕ = u² in R 3 , where λ > 0 and p ∈ [4, 6). Here, V (x) ∈ C ( R 3 , 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)]. [ABSTRACT FROM AUTHOR]