{
  "abstract": "We study the following prescribed scalar curvature problem $$ -Δu=K(y)u^{2^*-1},\\ \\ u>0 \\,\\ \\ \\text{in}\\ \\ \\mathbb{R}^N, $$ where $2^*=\\frac{2N}{N-2}$. For $N\\geq4$, we construct a positive $K(y)$ with maximum set $\\{|y|=1\\}$ and critical order of flatness $m=N-2$ at $|y|=1$, such that the equation admits only finitely many positive solutions in $D^{1,2}(\\mathbb{R}^N)$. To this end, we prove the compactness of the solution set for this equation by blow-up analysis and then apply Sard's theorem to choose $K$ for which each solution is non-degenerate and hence isolated, so that the solution set is finite. Our results also give a definite answer to Wei--Yan's conjecture in [J. Wei and S. Yan, JFA, 2010].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32707",
  "author_total": 2,
  "authors": [
    {
      "name": "Wei Ji"
    },
    {
      "name": "Yixing Pu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3j48ngyjayftx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32707",
  "fulltext_url": null,
  "id": "3j48ngyjayftx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32707",
  "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"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7223669386",
  "pdf_url": "https://arxiv.org/pdf/2610.32707",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3j48ngyjayftx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.32707",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32707"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Finitely many solutions for the prescribed scalar curvature problem at the critical flatness order",
  "title_text": "Finitely many solutions for the prescribed scalar curvature problem at the critical flatness order",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/3j48ngyjayftx",
  "venue": "arXiv",
  "versions": []
}