Positive and sign changing solutions to a nonlinear Choquard equation

Abstract

We consider the problem −Δu+W(x)u=(1|x|α∗|u|p)|u|p−2u,u∈H01(Ω), where Ω is an exterior domain in RN, N≥3,α∈(0,N), p∈[2,2N−αN−2),W∈C0(RN), infRNW>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.