{
  "abstract": "We prove that for every even integer $N\\geq 4$ or $N\\in\\{3,5,7\\}$, every axially symmetric solution to the $Q$-curvature-type problem $$ αP_N u + (N-1)!(1-\\frac{e^{Nu}}{\\int_{\\mathbb{S}^N} e^{Nu}dw})=0 \\ \\ \\ \\ \\ \\mbox{on} \\ \\mathbb{S}^N $$ is constant, provided that $ α\\ge\\frac{1}{2}$ and $α\\not =1$. The proof starts from a Gegenbauer expansion and its weighted $\\ell ^2$ estimate on coefficients. Then one key estimate is the refined estimate of the seminorm $\\lfloor G\\rfloor^2$ using a specially designed integration-by-parts identity for general even integer $N\\geq 4$. For $N\\in\\{3,5,7\\}$, the seminorm estimates are established via a descent to a three--dimensional nonlocal difference quotient identity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26227",
  "author_total": 4,
  "authors": [
    {
      "name": "Changfeng Gui"
    },
    {
      "name": "Tuoxin Li"
    },
    {
      "name": "Juncheng Wei"
    },
    {
      "name": "Zikai Ye"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2t4ewvb5gk2mc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26227",
  "fulltext_url": null,
  "id": "2t4ewvb5gk2mc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26227",
  "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": "W7214097336",
  "pdf_url": "https://arxiv.org/pdf/2609.26227",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2t4ewvb5gk2mc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26227",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26227"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Axially Symmetric Rigidity for a $Q$-Curvature-Type Equation on $ \\mathbb{S}^N $",
  "title_text": "Axially Symmetric Rigidity for a Q-Curvature-Type Equation on $ \\mathbb{S}^N $",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/2t4ewvb5gk2mc",
  "venue": "arXiv",
  "versions": []
}