Linear non-degeneracy and uniqueness of the bubble solution for the critical fractional Hénon equation in $ \mathbb{R}^N $

Abstract

We study the equation$\begin{equation*} (-\Delta)^s u = |x|^{\alpha} u^{\frac{N+2s+2\alpha}{N-2s}}\mbox{ in }\mathbb{R}^N, \end{equation*} \quad\quad\quad\quad\quad\quad\text{(P)}$where $ (-\Delta)^s $ is the fractional Laplacian operator with $ 0 < s < 1 $, $ \alpha>-2s $ and $ N>2s $. We prove the linear non-degeneracy of positive radially symmetric solutions of the equation (P) and, as a consequence, a uniqueness result of those solutions with Morse index equal to one. In particular, the ground state solution is unique. Our non-degeneracy result extends in the radial setting some known theorems done by Dávila, Del Pino and Sire (see [15, Theorem 1.1]), and Gladiali, Grossi and Neves (see [28, Theorem 1.3]).