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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 ofsolutions and lack variational information. Within the range of parameters ofMa-Zhao's result [Ma-Zhao, 2010] on symmetry, we prove a priori estimates forpositive solutions, generalizing the classical method of DeFigueiredo-Lions-Russbaum [De Figueiredo-Lions-Nussbaum, 1982] to the unboundeddomain and the nonlocal nonlinearity in our model. As an application, we showuniqueness and non-degeneracy results for the positive solution of the Choquardequation when $d \in \{ 3, 4, 5\}$, $p \ge 2$ and $(\alpha, p)$ close to $(d-2,2)$.

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