A uniqueness result for a Schrödinger-Poisson system with strong singularity

In this paper, we consider the following Schrödinger–Poisson system with strong singularity −∆u + φu = f(x)u , x ∈ Ω, −∆φ = u 2 , x ∈ Ω, u > 0, x ∈ Ω, u = φ = 0, x ∈ ∂Ω, where Ω ⊂ R3 is a smooth bounded domain, γ > 1, f ∈ L 1 (Ω) is a positive function (i.e. f(x) > 0 a.e. in Ω). A necessary and sufficient condition on the existence and uniqueness of positive weak solution of the system is obtained. The results supplement the main conclusions in recent literature