A priori estimates, uniqueness and nondegeneracy of positive solutions of the Choquard equation
Abstract
We consider the positive solutions for the nonlocal Choquard equation $ \Delta u + u  (\cdot^{\alpha} * u^p) u^{p2} 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 MaZhao's result [MaZhao, 2010] on symmetry, we prove a priori estimates for positive solutions, generalizing the classical method of De FigueiredoLionsRussbaum [De FigueiredoLionsNussbaum, 1982] to the unbounded domain and the nonlocal nonlinearity in our model. As an application, we show uniqueness and nondegeneracy results for the positive solution of the Choquard equation when $d \in \{ 3, 4, 5\}$, $p \ge 2$ and $(\alpha, p)$ close to $(d2, 2)$.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 arXiv:
 arXiv:2201.00368
 Bibcode:
 2022arXiv220100368L
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 39 pages