Uniqueness and Nondegeneracy of ground states of $ -\Delta u + (-\Delta)^s u+u = u^{p+1} \quad \hbox{in $\mathbb{R}^n$}$ when $s$ is close to $0$ and $1$

Abstract

We are concerned with the mixed local/nonlocal Schr\"{o}dinger equation \begin{equation} - \Delta u + (-\Delta)^s u+u = u^{p+1} \quad \hbox{in $\mathbb{R}^n$,} \end{equation} for arbitrary space dimension $n\geqslant1$, $s\in(0,1)$, and $p\in(0,2^*-2)$ with $2^*$ the critical Sobolev exponent. We provide the existence and several fundamental properties of nonnegative solutions for the above equation. And then, we prove that, if $s$ is close to $0$ and $1$, respectively, such equation then possesses a unique (up to translations) ground state, which is nondegenerate.