Positive and sign changing solutions to a nonlinear Choquard equation

We consider the problem -Delta u + W(x)u = (1/vertical bar x vertical bar(alpha) * vertical bar u vertical bar(p)) vertical bar u vertical bar(p-2)u, u is an element of H-0(1)(Omega), where Omega is an exterior domain in R-N, N >= 3, alpha is an element of (0, N), p is an element of [2, 2N-alpha/N-2), W is an element of C-0(R-N), inf(RN) W > 0, and W(x) -> V-infinity -> 0 as vertical bar x vertical bar -> infinity. Under symmetry assumptions on Omega 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. (C) 2013 Elsevier Inc. All rights reserved.