The m-Laplace equation with a gradient term I: Liouville theorem in the second subcritical case
Abstract
Let 1<m<n, q>0, and p∈ℝ. We study positive C1loc weak solutions of −Δm u=up|Du|q in ℝn. If 0<q<m−1, we prove that every such solution is constant provided p+q−m+1<((m−1)(m−q)2)/((n−m)(m−1−q)). If qgeqslant m−1, the same conclusion holds for every p∈ℝ. For 0<q<m, the proof uses an auxiliary function and reduces the main differential estimate to a lower bound for a function of one variable. An explicit construction and a local maximum-principle argument then prove constancy without differentiating across {|Du|=0}. When m=2, our result answers the natural Liouville problem left open by Bidaut-Véron, García-Huidobro and Véron for the whole range 0<q<1. The case q=m is handled by an increasing change of the dependent variable, in the spirit of the related work of Bidaut-Véron, while the case q>m follows from the theorem of Lu and Zhu.
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BibTeX
@misc{wu2026mlaplace,
title = {{The $m$-Laplace equation with a gradient term I: Liouville theorem in the second subcritical case}},
author = {Tian Wu and Jin Yan and Hua Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30006},
eprint = {2609.30006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30006},
}
RIS
TY - UNPB
TI - The m-Laplace equation with a gradient term I: Liouville theorem in the second subcritical case
AU - Tian Wu
AU - Jin Yan
AU - Hua Zhu
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30006
UR - https://arxiv.org/abs/2609.30006
AB - Let 1<m<n, q>0, and p∈ℝ. We study positive C^1_loc weak solutions of −Δ_m u=u^p|Du|^q in ℝ^n. If 0<q<m−1, we prove that every such solution is constant provided p+q−m+1<((m−1)(m−q)^2)/((n−m)(m−1−q)). If qgeqslant m−1, the same conclusion holds for every p∈ℝ. For 0<q<m, the proof uses an auxiliary function and reduces the main differential estimate to a lower bound for a function of one variable. An explicit construction and a local maximum-principle argument then prove constancy without differentiating across {|Du|=0}. When m=2, our result answers the natural Liouville problem left open by Bidaut-Véron, García-Huidobro and Véron for the whole range 0<q<1. The case q=m is handled by an increasing change of the dependent variable, in the spirit of the related work of Bidaut-Véron, while the case q>m follows from the theorem of Lu and Zhu.
ER -