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

Zexing Li

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arXiv · 2201.00368

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

Zexing Li

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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 $(d-2, 2)$.

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Zexing Li. 2022-01-02. A priori estimates, uniqueness and non-degeneracy of positive solutions of the Choquard equation. https://arxiv.org/abs/2201.00368

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