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

TL;DR

This paper investigates the existence, regularity, symmetry, and decay properties of groundstate solutions to a class of nonlinear Choquard equations involving Riesz potentials, providing comprehensive qualitative analysis.

Contribution

It establishes the existence of positive groundstates, proves their regularity and radial symmetry, and derives their decay asymptotics, extending understanding of nonlinear Choquard equations.

Findings

Existence of positive groundstate solutions for certain parameter ranges.

Groundstates are regular, positive, radially symmetric, and monotone decreasing.

Decay asymptotics of groundstates at infinity are characterized.

Abstract

We consider a semilinear elliptic problem [- \Delta u + u = (I_\alpha \ast \abs{u}^p) \abs{u}^{p - 2} u \quad\text{in (\mathbb{R}^N),}] where (I_\alpha) is a Riesz potential and (p>1). This family of equations includes the Choquard or nonlinear Schr\"odinger-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.