Critical anisotropic p-Laplace equations without finite energy: classification and a Schoen-type Harnack inequality
Abstract
Let N≥ 2 and 1<p<N. Let H be a possibly non-reversible Minkowski norm on ℝN, with H∈ C2(ℝN∖{0}) and D2(H2/2) uniformly positive definite away from the origin. We classify all nonnegative local weak solutions u∈ W1,ploc(ℝN) of −div(H(∇ u)p−1∇ H(∇ u))=up*−1, p*=Np/(N−p), without any finite-energy, integrability, or asymptotic assumption. Every nonzero solution is an anisotropic Sobolev bubble. This settles, for the full range 1<p<N, the anisotropic analogue of Problem A of Antonini--Ciraolo--Gatti, whose Euclidean case was settled by Zhang, and removes the restriction on p in the non-energy anisotropic classification of Chen--Wu--Yan--Yang. We also prove a scale-invariant Schoen-type Harnack inequality for local solutions. The proof follows the strategy of Zhang for the Euclidean p-Laplacian, combining rigidity of bounded blow-up profiles and a first-contact argument with the integral bootstrap of Ciraolo and Gatti. The anisotropic analysis requires two additional ingredients. First, we establish a Bochner identity for the conjugated stress tensor in distributional form on the noncritical set, using only C2 regularity of H. Second, we prove a two-term expansion at isolated singularities of Finsler p-harmonic functions, including the gradient control needed by the Pohozaev invariant. For non-reversible anisotropies the argument is formulated with the reflected norm. A Morrey-type estimate proved directly for the relevant Leray--Lions operator replaces the potential-theoretic input used in the Euclidean argument.
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BibTeX
@misc{le2026critical,
title = {{Critical anisotropic $p$-Laplace equations without finite energy: classification and a Schoen-type Harnack inequality}},
author = {Phuong Le},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00337},
eprint = {2610.00337},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00337},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Critical anisotropic p-Laplace equations without finite energy: classification and a Schoen-type Harnack inequality
AU - Phuong Le
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00337
UR - https://arxiv.org/abs/2610.00337
AB - Let N≥ 2 and 1<p<N. Let H be a possibly non-reversible Minkowski norm on ℝ^N, with H∈ C^2(ℝ^N∖{0}) and D^2(H^2/2) uniformly positive definite away from the origin. We classify all nonnegative local weak solutions u∈ W^1,p_loc(ℝ^N) of −div(H(∇ u)^p−1∇ H(∇ u))=u^p^*−1, p^*=Np/(N−p), without any finite-energy, integrability, or asymptotic assumption. Every nonzero solution is an anisotropic Sobolev bubble. This settles, for the full range 1<p<N, the anisotropic analogue of Problem A of Antonini--Ciraolo--Gatti, whose Euclidean case was settled by Zhang, and removes the restriction on p in the non-energy anisotropic classification of Chen--Wu--Yan--Yang. We also prove a scale-invariant Schoen-type Harnack inequality for local solutions. The proof follows the strategy of Zhang for the Euclidean p-Laplacian, combining rigidity of bounded blow-up profiles and a first-contact argument with the integral bootstrap of Ciraolo and Gatti. The anisotropic analysis requires two additional ingredients. First, we establish a Bochner identity for the conjugated stress tensor in distributional form on the noncritical set, using only C^2 regularity of H. Second, we prove a two-term expansion at isolated singularities of Finsler p-harmonic functions, including the gradient control needed by the Pohozaev invariant. For non-reversible anisotropies the argument is formulated with the reflected norm. A Morrey-type estimate proved directly for the relevant Leray--Lions operator replaces the potential-theoretic input used in the Euclidean argument.
ER -