A priori estimates, uniqueness and non-degeneracy of positive solutions of the Choquard equation

TL;DR

This paper establishes a priori estimates, uniqueness, and non-degeneracy of positive solutions to the nonlocal Choquard equation in certain parameter regimes, extending classical methods to unbounded domains and nonlocal nonlinearities.

Contribution

It generalizes classical a priori estimate methods to the nonlocal Choquard equation on unbounded domains, providing new uniqueness and non-degeneracy results for positive solutions.

Findings

A priori estimates for positive solutions in the parameter range of Ma-Zhao's symmetry results.

Proof of uniqueness of positive solutions for specific dimensions and parameters.

Demonstration of non-degeneracy of solutions near the critical parameter values.

Abstract

We consider the positive solutions for the nonlocal Choquard equation $- \Delta u + u - (|\cdot|^{-\alpha} * |u|^p) |u|^{p-2} u = 0$ in $\mathbb{R}^d$. Compared with ground states, positive solutions form a larger class of solutions and lack variational information. Within the range of parameters of Ma-Zhao's result [Ma-Zhao, 2010] on symmetry, we prove a priori estimates for positive solutions, generalizing the classical method of De Figueiredo-Lions-Russbaum [De Figueiredo-Lions-Nussbaum, 1982] to the unbounded domain and the nonlocal nonlinearity in our model. As an application, we show uniqueness and non-degeneracy results for the positive solution of the Choquard equation when $d \in \{ 3, 4, 5\}$, $p \ge 2$ and $(\alpha, p)$ close to $(d-2, 2)$.