Existence, uniqueness and multiplicity of positive solutions for Schrödinger–Poisson system with singularity

In this paper, we consider the following Schrödinger–Poisson system with singularity{−Δu+ηϕu=μu−r,inΩ,−Δϕ=u2,inΩ,u>0,inΩ,u=ϕ=0,on∂Ω, where Ω⊂R3Ω⊂R3 is a smooth bounded domain with boundary ∂Ω, η=±1η=±1, r∈(0,1)r∈(0,1) is a constant, μ>0μ>0 is a parameter. We obtain the existence and uniqueness of positive solution for η=1η=1 and any μ>0μ>0 by using the variational method. The existence and multiplicity of solutions for the system are also considered for η=−1η=−1 and μ>0μ>0 small enough by using the method of Nehari manifold.