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The m-Laplace equation with a gradient term II: classification in the second critical case
Abstract
Let 1<m<n and 0<q<m−1. We classify positive, locally bounded weak solutions of −Δm u=up|Du|q in ℝn, p=(m−q)/(n−m)(n+q/(m−1−q))−1. Every solution is constant or belongs to an explicit family of radial solutions, up to translation and scaling. No global bound, decay, or finite-energy assumption is required. The proof is based on a scalar differential identity and a maximum principle that remains valid at critical points. These lead to a sharp gradient bound, classification of normalized solutions, and a first-contact argument for arbitrary entire solutions. Local regularity and the behavior at critical points are treated throughout in the weak solution class; global C2 regularity is not assumed.
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BibTeX
@misc{wu2026mlaplace,
title = {{The $m$-Laplace equation with a gradient term II: classification in the second critical case}},
author = {Tian Wu and Jin Yan and Hua Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33583},
eprint = {2609.33583},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33583},
}
RIS
TY - UNPB
TI - The m-Laplace equation with a gradient term II: classification in the second critical case
AU - Tian Wu
AU - Jin Yan
AU - Hua Zhu
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33583
UR - https://arxiv.org/abs/2609.33583
AB - Let 1<m<n and 0<q<m−1. We classify positive, locally bounded weak solutions of −Δ_m u=u^p|Du|^q in ℝ^n, p=(m−q)/(n−m)(n+q/(m−1−q))−1. Every solution is constant or belongs to an explicit family of radial solutions, up to translation and scaling. No global bound, decay, or finite-energy assumption is required. The proof is based on a scalar differential identity and a maximum principle that remains valid at critical points. These lead to a sharp gradient bound, classification of normalized solutions, and a first-contact argument for arbitrary entire solutions. Local regularity and the behavior at critical points are treated throughout in the weak solution class; global C^2 regularity is not assumed.
ER -