Schrödinger–Poisson system with singular potential

Consider the following Schrödinger–Poisson system(SP){−Δu+Vλ(x)u+ϕ(x)u=|u|p−1u,x=(y,z)∈R2×R,−Δϕ=u2,lim|x|→+∞ϕ(x)=0, where Vλ=λ+1|y|α with λ≥0λ≥0, y=(x1,x2)∈R2y=(x1,x2)∈R2 and |y|=x12+x22. When α∈[0,8)α∈[0,8) and max{2,2+α2}