Nondegeneracy of the bubble for the critical p-Laplace equation

Abstract

We prove the non-degeneracy of the extremals of the Sobolev inequality $$\underset{{\mathbb{R}}^N}{\int }|\nabla u{|}^{p}dx\ge {\mathcal{S}}_p\underset{{\mathbb{R}}^N}{\int }|u{|}^{\frac{Np}{N-p}}dx,u\in {\mathcal{D}}^{1,p}({\mathbb{R}}^N)$$ when $1<p<N,$ as solutions of a critical quasilinear equation involving the $p-$Laplacian.