Uniqueness and asymptotical behavior of solutions to a Choquard equation with singularity

Abstract In this paper, we consider a nonautonomous Choquard equation with singularity − Δ u + V ( x ) u + λ ( I α ∗ | u | p ) | u | p − 2 u = f ( x ) u − γ , x ∈ R 3 , u > 0 , x ∈ R 3 , where I α is the Riesz potential of order α ∈ ( 0 , 3 ) and 1 + α 3 ≤ p 3 + α , 0 γ 1 . Under certain assumptions on V and f , we show the existence and uniqueness of positive solution for λ > 0 by using variational method. We also study the asymptotic behavior of solutions as λ → 0 .