Unique continuation, nonexistence and bubbling for the critical p-Laplace equation in the plane
Abstract
For 1<p<2 we prove weak and strong unique continuation properties for the solutions to Δpu+f(u)=0 in a planar domain, with f continuous and such that |f(s)|≤ C|s|p−1: a solution vanishing on an open set vanishes identically, and so does a solution vanishing to infinite order at a single point. We can therefore make progress towards a proof of some long-standing nonexistence results available only for p=2, such as a celebrated one of Esteban and Lions, establishing here that for the critical p-Laplace equation with zero Dirichlet boundary condition on a half-plane, there are no nontrivial finite energy solutions. In the plane, unique continuation thus provides the missing ingredient for a generalisation to the p-Laplacian operator of a classical result of Struwe on the bubble-profile decomposition of possibly sign-changing Palais-Smale sequences associated to the Brezis-Nirenberg problem, which remains open in dimension N≥3 for p≠2. We obtain a characterisation of their loss of compactness in terms of the finite energy solutions of Δpu+|u|p*−2u=0 in ℝ2.
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BibTeX
@misc{mercuri2026unique,
title = {{Unique continuation, nonexistence and bubbling for the critical $p$-Laplace equation in the plane}},
author = {Carlo Mercuri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.34204},
eprint = {2609.34204},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34204},
}
RIS
TY - UNPB
TI - Unique continuation, nonexistence and bubbling for the critical p-Laplace equation in the plane
AU - Carlo Mercuri
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2609.34204
UR - https://arxiv.org/abs/2609.34204
AB - For 1<p<2 we prove weak and strong unique continuation properties for the solutions to Δ_pu+f(u)=0 in a planar domain, with f continuous and such that |f(s)|≤ C|s|^p−1: a solution vanishing on an open set vanishes identically, and so does a solution vanishing to infinite order at a single point. We can therefore make progress towards a proof of some long-standing nonexistence results available only for p=2, such as a celebrated one of Esteban and Lions, establishing here that for the critical p-Laplace equation with zero Dirichlet boundary condition on a half-plane, there are no nontrivial finite energy solutions. In the plane, unique continuation thus provides the missing ingredient for a generalisation to the p-Laplacian operator of a classical result of Struwe on the bubble-profile decomposition of possibly sign-changing Palais-Smale sequences associated to the Brezis-Nirenberg problem, which remains open in dimension N≥3 for p≠2. We obtain a characterisation of their loss of compactness in terms of the finite energy solutions of Δ_pu+|u|^p^*−2u=0 in ℝ^2.
ER -