{
  "abstract": "Let $n\\geq2$, $0<β<n$, and let $V$ be a continuous nonnegative potential on $\\R^n$ with a positive lower bound outside a ball. Under the constraint $\\int_{\\R^n}(|\\nabla u|^n+V|u|^n)\\dd\\leq1$, we prove the singular Moser--Trudinger inequality with sharp coefficient $(1-β/n)α_n$, where $α_n=n|\\mathbb S^{n-1}|^{1/(n-1)}$. The subcritical supremum is attained for every such potential. If $V$ is radial and $r^βV(r)$ is nondecreasing, the critical supremum is also attained. This includes potentials that vanish on a ball and decrease to a positive limit at infinity. Under the same radial hypothesis, a bound for the tangential gradient shows that every extremal is radial and that the tangential gradients of maximizing sequences converge to zero in $L^n$, including at the critical coefficient. For the singular $n$-Laplacian equation with critical exponential growth, we obtain an explicit sufficient level for Palais--Smale compactness. Under the Ambrosetti--Rabinowitz condition and a smallness assumption at zero, this gives a nonzero nonnegative mountain-pass solution for all sufficiently large coupling parameters.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07628",
  "author_total": 3,
  "authors": [
    {
      "name": "Hao Hu"
    },
    {
      "name": "Yanjun Liu"
    },
    {
      "name": "Huabo Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5h9f39g9v5ebn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07628",
  "fulltext_url": null,
  "id": "5h9f39g9v5ebn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07628",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07628",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5h9f39g9v5ebn.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07628",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07628"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp singular Moser-Trudinger inequalities and n-Laplacian equations with degenerate potentials",
  "title_text": "Sharp singular Moser-Trudinger inequalities and n-Laplacian equations with degenerate potentials",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/5h9f39g9v5ebn",
  "venue": "arXiv",
  "versions": []
}