Schrödinger–Poisson system with singular potential

Consider the following Schrödinger–Poisson system ( SP ) { − Δ u + V λ ( x ) u + ϕ ( x ) u = | u | p − 1 u , x = ( y , z ) ∈ R 2 × R , − Δ ϕ = u 2 , lim | x | → + ∞ ϕ ( x ) = 0 , where V λ = λ + 1 | y | α with λ ≥ 0 , y = ( x 1 , x 2 ) ∈ R 2 and | y | = x 1 2 + x 2 2 . When α ∈ [ 0 , 8 ) and max { 2 , 2 + α 2 } < p < 5 , the existence and a priori estimate of positive solutions of problem (SP) are established in suitable weighted Sobolev space. Moreover, the asymptotic behavior of the solutions as λ → 0 is also discussed.