Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics

We consider a semilinear elliptic problem -Delta u+u = (I alpha*vertical bar u vertical bar(p))vertical bar u vertical bar(p-2)u R-N, where I-alpha is a Riesz potential and p > 1. This family of equations includes the Choquard or nonlinear Schrodinger Newton equation. For an optimal range of parameters we prove the existence of a positive groundstate solution of the equation. We also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point. Finally, we derive the decay asymptotics at infinity of the groundstates. (C) 2013 Elsevier Inc. All rights reserved.