Uniqueness and nondegeneracy of solutions to $(-Δ)^su+ u=u^p$ in $\mathbb{R}^N$ and in balls

Mouhamed Moustapha Fall, Tobias Weth

Published 2023-10-16Version 1

We prove that positive solutions $u\in H^s(\mathbb{R}^N)$ to the equation $(-\Delta )^s u+ u=u^p$ in $\mathbb{R}^N$ are unique up to translations and nondegenerate, for all $s\in (0,1)$, $N\geq 1$ and $p>1$ is strictly smaller than the critical Sobolev exponent. This generalizes a result of Frank, Lenzmann and Silvestre \cite{FLS}, where the same uniqueness and nondegeneracy is proved for solutions with Morse index 1. Letting $B$ be the unit centered ball and $\lambda_1(B)$ the first Dirichlet eigenvalue of the fractional Laplacian $(-\Delta )^s$, we also prove that positive solutions to $(-\Delta )^s u+\lambda u=u^p$ in ${B}$ with $u=0$ on $\mathbb{R}^N\setminus B$, are unique and nondegenerate for any $\lambda> -\lambda_1(B)$. This extends the very recent results in \cite{DIS-1,Azahara-Parini} which provide uniqueness and nondegeneracy of least energy solutions.