{
  "abstract": "We study the equation \\[ -Δ_{m,g}u=λe^u|\\nabla u|^q \\] on complete, connected, noncompact Riemannian manifolds of dimension $n\\ge2$ with $\\Ric_g\\ge0$. For every $m>1$, $q\\in\\mathbb R$ and $λ>0$, we prove that every positive $C^1$ weak solution is constant when $q>0$, while no such solution exists when $q\\le0$. For $q>m$, we derive a local gradient estimate independent of the solution by Nash--Moser iteration, which leads to automatic boundedness and resolves the question raised by Lu and Zhu~\\cite[Remark~1.6(B)]{LZ}. For $m-1<q<m$, we use an auxiliary function and a divergence inequality to extend the Liouville result beyond the range covered by existing gradient-inequality methods. The remaining cases are treated by suitable changes of variables and integral estimates. The conclusions also extend to real-valued solutions bounded below.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23029",
  "author_total": 3,
  "authors": [
    {
      "name": "Yuhua Sun"
    },
    {
      "name": "Zhen Wang"
    },
    {
      "name": "Mengyu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7cxeg4bw45qn6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23029",
  "fulltext_url": null,
  "id": "7cxeg4bw45qn6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23029",
  "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": "W7222916645",
  "pdf_url": "https://arxiv.org/pdf/2610.23029",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7cxeg4bw45qn6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23029",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23029"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Liouville theorems for $-Δ_{m,g}u=λe^u|\\nabla u|^q$ in the full range of gradient exponents",
  "title_text": "Liouville theorems for −Δ_m,gu=λe^u|∇ u|^q in the full range of gradient exponents",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/7cxeg4bw45qn6",
  "venue": "arXiv",
  "versions": []
}