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

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 $ is fractional Laplacian operator with 0 < s 1 $, \alpha>-2s and N>2s $. prove linear non-degeneracy of positive radially symmetric solutions equation (P) and, as a consequence, uniqueness result those Morse index equal to one. In particular, ground state solution unique. Our extends radial setting some known theorems done by Dávila, Del Pino Sire (see [15, Theorem 1.1]), Gladiali, Grossi Neves [28, 1.3]).