The complete spectrum of the linearized $p$-Laplacian at a Sobolev extremal

Abstract

Let $1<p<n$ and let $v(x)=(1+|x{|}^{p/(p-1)}{)}^{-(n-p)/p}$ be the standard radial extremal for the sharp Sobolev inequality. We determine all eigenvalues and eigenspaces of the linearized $p$-Laplacian at $v$, defined by its closed quadratic form in $L^2({\mathbb{R}}^n,v^{p^{\ast }-2}dx)$. After decomposition into spherical harmonics, an explicit gauge transformation and a change of variables identify each radial operator with a shifted Jacobi operator. This yields a complete eigenbasis indexed by $(\ell ,k)\in {\mathbb{N}}_0^2$, where $\ell$ is the angular degree and $k$ is the radial mode number.