Uniqueness and nondegeneracy of positive solutions of $\Ds u+u=u^p$ in $\R^N$ when $s$ is close to 1

Abstract: We consider the equation $\Ds u+u=u^p$, with $s\in(0,1)$ in the subcritical range of $p$. We prove that if $s$ is sufficiently close to 1 the equation possesses a unique minimizer, which is nondegenerate.