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

Abstract

We consider the positive solutions for the nonlocal Choquard equation $-\Delta u+u-(|\cdot {|}^{-\alpha }\ast |u{|}^p)|u{|}^{p-2}u=0$ 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 $(d-2,2)$.

39 pages