Positive and sign changing solutions to a nonlinear Choquard equation

We consider the problem −Δu+W(x)u=(1|x|α∗|u|p)|u|p−2u,u∈H01(Ω), where ΩΩ is an exterior domain in RNRN, N≥3N≥3,α∈(0,N)α∈(0,N), p∈[2,2N−αN−2),W∈C0(RN), infRNW>0infRNW>0, and W(x)→V∞>0W(x)→V∞>0 as |x|→∞|x|→∞. Under symmetry assumptions on ΩΩ and WW, which allow finite symmetries, and some assumptions on the decay of WW at infinity, we establish the existence of a positive solution and multiple sign changing solutions to this problem, having small energy.