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*}\label{P0} (-Δ)^s u = |x|^α u^{\frac{N+2s+2α}{N-2s}}\mbox{ in }\mathbb{R}^N,\tag{P} \end{equation*} 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 (\ref{P0}) 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 \cite[Theorem 1.1]{Davila-DelPino-Sire}), and Gladiali, Grossi and Neves (see \cite[Theorem 1.3]{Gladiali-Grossi-Neves}).

References in corpus (2)

• Nonradial solutions for the Hénon equation in $R^N$
• Non degeneracy of the bubble in the critical case for non local equations