Uniqueness and Nondegeneracy of Ground States for Choquard Equations in three dimensions

We obtain uniqueness and nondegeneracy results for ground states of Choquard equations $-\mathrm{\Delta }u+u=(\mathrm{\mid }x{\mathrm{\mid }}^{-1}\ast \mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u$ in ${\mathbb{R}}^3$, provided that $p>2$ and $p$ is sufficiently close to 2.