{
  "abstract": "Let $1<m<n$, $q>0$, and $p\\in\\mathbb R$. We study positive $C^1_{\\mathrm{loc}}$ weak solutions of \\[-Δ_m u=u^p|Du|^q\\qquad\\text{in }\\mathbb R^n.\\] If $0<q<m-1$, we prove that every such solution is constant provided \\[p+q-m+1<\\frac{(m-1)(m-q)^2}{(n-m)(m-1-q)}.\\] If $q\\geqslant m-1$, the same conclusion holds for every $p\\in\\mathbb R$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30006",
  "author_total": 3,
  "authors": [
    {
      "name": "Tian Wu"
    },
    {
      "name": "Jin Yan"
    },
    {
      "name": "Hua Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7vwgkh4m0yfqg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30006",
  "fulltext_url": null,
  "id": "7vwgkh4m0yfqg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30006",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214279550",
  "pdf_url": "https://arxiv.org/pdf/2609.30006",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7vwgkh4m0yfqg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.30006",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30006"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The $m$-Laplace equation with a gradient term I: Liouville theorem in the second subcritical case",
  "title_text": "The m-Laplace equation with a gradient term I: Liouville theorem in the second subcritical case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/7vwgkh4m0yfqg",
  "venue": "arXiv",
  "versions": []
}