{
  "abstract": "We study stable solutions of the mean curvature equation $$ -\\operatorname{div}\\left(\\frac{\\nabla u}{\\sqrt{1+|\\nabla u|^2}}\\right) =|u|^{p-1}u $$ in domains of $\\mathbb R^n$. We establish Liouville-type results in three settings. First, for globally stable solutions in $\\mathbb R^n$, we prove that $u\\equiv0$ if $1<p<p_{JL}$, where $p_{JL}$ is the Joseph--Lundgren exponent. Second, for solutions in $\\mathbb R^n$ that are stable outside a compact set, we show that if $n>2$ and $p=p_S$, where $p_S$ is the Sobolev critical exponent, then $u\\equiv0$. Third, for solutions of the Dirichlet problem on connected unbounded domains with smooth boundary satisfying an epigraph-type geometric condition, we prove that if $n>2$, $1<p\\le p_S$, and the solution is stable outside a compact set, then $u\\equiv0$. The proofs are based on a gradient estimate for stable solutions, an integrability result, conservation laws and Pohozaev-type identities, and a unique continuation argument. These results extend Liouville-type theorems for stable solutions from the semilinear Lane--Emden equation to the mean curvature equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14652",
  "author_total": 1,
  "authors": [
    {
      "name": "Fanheng Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dp2xn3vwnwt87.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14652",
  "fulltext_url": null,
  "id": "dp2xn3vwnwt87",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14652",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7221827609",
  "pdf_url": "https://arxiv.org/pdf/2610.14652",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dp2xn3vwnwt87.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14652",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14652"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Liouville theorems for stable solutions to the mean curvature equation",
  "title_text": "Liouville theorems for stable solutions to the mean curvature equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/dp2xn3vwnwt87",
  "venue": "arXiv",
  "versions": []
}