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

• Published in: Journal of functional analysis Vol. 265; no. 2; pp. 153 - 184
• Main Authors: Moroz, Vitaly, Van Schaftingen, Jean
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.07.2013
• Subjects: Decay asymptotics; Existence; Nonlocal semilinear elliptic problem; Pohožaev identity; Riesz potential; Stationary Choquard equation; Stationary Hartree equation; Stationary nonlinear Schrödinger–Newton equation; Symmetry; Pohožaev identity; Stationary nonlinear Schrödinger–Newton equation; Stationary Hartree equation; Decay asymptotics; Nonlocal semilinear elliptic problem; Stationary Choquard equation; Riesz potential; Existence; Symmetry
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2013.04.007

We consider a semilinear elliptic problem−Δu+u=(Iα⁎|u|p)|u|p−2uinRN, where Iα is a Riesz potential and p>1. This family of equations includes the Choquard or nonlinear Schrödinger–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.