{
  "abstract": "Let $N=m+1$ and consider the thin ellipsoid \\[ Ω_\\varepsilon=\\{(y,x_N)\\in\\mathbb R^m\\times\\mathbb R:\\ |y|^2+\\varepsilon^{-2}x_N^2<1\\}. \\] We prove that in every sufficiently large dimension, for every sufficiently small $\\varepsilon>0$, there exists a smooth positive, strictly increasing, strictly convex, superlinear nonlinearity $f_\\varepsilon$ for which the extremal solution in $Ω_\\varepsilon$ is an unbounded $H^1_0$ solution. Combining this result with Dancer's thin-domain regularity theorem for the Gelfand nonlinearity $f(t)=e^t$, we obtain on the same sufficiently thin ellipsoids a bounded Gelfand extremal solution and an unbounded extremal solution for another nonlinearity. Thus, in this two-part sense, Brézis' Open Problem~6.1 is resolved in every sufficiently large dimension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06673",
  "author_total": 2,
  "authors": [
    {
      "name": "Bao Yu"
    },
    {
      "name": "Yang Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a9qxv56t7t2y7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06673",
  "fulltext_url": null,
  "id": "a9qxv56t7t2y7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06673",
  "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": "W7212087010",
  "pdf_url": "https://arxiv.org/pdf/2609.06673",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a9qxv56t7t2y7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06673",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06673"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Singular Extremal Solutions on Thin Ellipsoids with Varying Nonlinearities",
  "title_text": "Singular Extremal Solutions on Thin Ellipsoids with Varying Nonlinearities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/a9qxv56t7t2y7",
  "venue": "arXiv",
  "versions": []
}