# Nondegeneracy of the bubble for the critical p-Laplace equation

## TL;DR

This paper proves the non-degeneracy of extremal functions for the Sobolev inequality involving the p-Laplacian, which is crucial for understanding the stability and uniqueness of solutions in critical quasilinear PDEs.

## Contribution

It establishes the non-degeneracy of extremals for the Sobolev inequality with the p-Laplacian, a key step in analyzing the structure of solutions to related PDEs.

## Key findings

- Proves non-degeneracy of extremals for the p-Laplacian Sobolev inequality.
- Provides a foundation for stability analysis of solutions.
- Enhances understanding of critical quasilinear equations.

## Abstract

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