PreprintarXiv licencearXiv
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 L2(ℝn,vp*−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 (ℓ,k)∈ℕ02, where ℓ is the angular degree and k is the radial mode number.
Related
- Sobolev Minimizers for Critical Vectorial p-Laplacian Problems
- Extremal and uniqueness results for a singular p-Laplacian Caputo-Hadamard fractional equation involving parameter dependence
- Sharp Born--Infeld Sobolev inequality and extremal functions on the lattice graph ℤN
- Eigenvalue comparison theorems for the Witten-Laplacian and the weighted p-Laplacian on complete manifolds with a modified Ricci curvature bounded from below
- The homogeneity function of the fractional p-Laplacian and optimal asymptotic exponents for critical Hardy-Sobolev equations
BibTeX
@misc{zhang2026complete,
title = {{The complete spectrum of the linearized $p$-Laplacian at a Sobolev extremal}},
author = {Yitian Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27276},
eprint = {2608.27276},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27276},
}
RIS
TY - UNPB
TI - The complete spectrum of the linearized p-Laplacian at a Sobolev extremal
AU - Yitian Zhang
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.27276
UR - https://arxiv.org/abs/2608.27276
AB - 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(ℝ^n,v^p^*−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 (ℓ,k)∈ℕ_0^2, where ℓ is the angular degree and k is the radial mode number.
ER -