Unique continuation, nonexistence and bubbling for the critical p-Laplace equation II: the degenerate elliptic case and higher dimensions
Abstract
For every p>1, we prove that a nontrivial solution of −Δpu=f(x,u,∇ u) in a planar domain, with |f(x,s,ξ)|≤ C(|s|p−1+|ξ|p−1), vanishes neither on an open set nor to infinite order at a point. This allows us to prove unique continuation properties of axially symmetric solutions to p-Laplace equations in higher dimensions N≥3, with axially symmetric p-harmonic functions as a particular case. Unlike in our previous work, the analysis is based here on the regularity and quasiconformal properties of the whole stress field |∇ u|p−2∇ u in the plane, taking into account the more delicate regularity theory for p>2. This allows us to prove Liouville-type and bubble profile decomposition results for critical problems in dimensions N≥3 and in the whole range 1<p<N, under some symmetry conditions.
Related
- Unique continuation, nonexistence and bubbling for the critical p-Laplace equation in the plane
- Sign-changing bubbling solutions on polygons for the Brezis-Nirenberg problem in high dimensions
- The fractional Hénon-Lane-Emden conjecture and some critical nonexistence phenomena
- Well-posedness of the two-dimensional unsteady Prandtl system in Sobolev space with degenerate critical points
- A class of degenerate elliptic problems involving the Grushin operator: critical mass and ground states
BibTeX
@misc{mercuri2026unique,
title = {{Unique continuation, nonexistence and bubbling for the critical $p$-Laplace equation II: the degenerate elliptic case and higher dimensions}},
author = {Carlo Mercuri},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06694},
eprint = {2611.06694},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06694},
}
RIS
TY - UNPB
TI - Unique continuation, nonexistence and bubbling for the critical p-Laplace equation II: the degenerate elliptic case and higher dimensions
AU - Carlo Mercuri
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06694
UR - https://arxiv.org/abs/2611.06694
AB - For every p>1, we prove that a nontrivial solution of −Δ_pu=f(x,u,∇ u) in a planar domain, with |f(x,s,ξ)|≤ C(|s|^p−1+|ξ|^p−1), vanishes neither on an open set nor to infinite order at a point. This allows us to prove unique continuation properties of axially symmetric solutions to p-Laplace equations in higher dimensions N≥3, with axially symmetric p-harmonic functions as a particular case. Unlike in our previous work, the analysis is based here on the regularity and quasiconformal properties of the whole stress field |∇ u|^p−2∇ u in the plane, taking into account the more delicate regularity theory for p>2. This allows us to prove Liouville-type and bubble profile decomposition results for critical problems in dimensions N≥3 and in the whole range 1<p<N, under some symmetry conditions.
ER -