Linear non-degeneracy and uniqueness of the bubble solution for the critical fractional Hénon equation in ℝ^N

2020/09/20 by Alarcón, S., Barrios, B., Quaas, A.
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2009.09481

Abstract

We study the equation (-Δ)^s u = |x|^α u^{(N+2s+2α)/(N-2s)} in ℝ^N, where (-Δ)^s is the fractional Laplacian operator with 0 < s < 1, α>-2s and N>2s. We prove the linear non-degeneracy of positive radially symmetric solutions of the equation (\refP0) 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).

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