Positive and sign changing solutions to a nonlinear Choquard equation

We consider the problem − Δ u + W ( x ) u = ( 1 | x | α ∗ | u | p ) | u | p − 2 u , u ∈ H 0 1 ( Ω ) , where Ω is an exterior domain in R N , N ≥ 3 , α ∈ ( 0 , N ) , p ∈ [ 2 , 2 N − α N − 2 ) , W ∈ C 0 ( R N ) , inf R N W > 0 , and W ( x ) → V ∞ > 0 as | x | → ∞ . Under symmetry assumptions on Ω and W , which allow finite symmetries, and some assumptions on the decay of W at infinity, we establish the existence of a positive solution and multiple sign changing solutions to this problem, having small energy.