{
  "abstract": "We establish a quantitative Gidas--Ni--Nirenberg theorem for the pure fractional Dirichlet problem \\[ (-Δ)^s u=k(x)g(u)\\quad\\text{in }B_1, \\qquad u=0\\quad\\text{in }\\mathbb R^n\\setminus B_1, \\qquad 0<s<1. \\] For positive bounded solutions under a two-sided $L^\\infty$ normalization, we prove \\[ \\mathcal D(u)\\le C\\,\\mathcal D(k)^γ,\\qquad 0<γ<1, \\] where $\\mathcal D$ measures spherical oscillation and outward radial increase. To the best of our knowledge, this provides the first quantitative Gidas--Ni--Nirenberg estimate for the pure fractional Dirichlet problem. The key new ingredient is a shift-uniform, weak Harnack inequality for shifted antisymmetric supersolutions, which overcomes the exterior-tail obstruction in the quantitative moving-plane argument. This mechanism extends to nonseparable nonlinearities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14618",
  "author_total": 2,
  "authors": [
    {
      "name": "Qizhen Shen"
    },
    {
      "name": "Jiongduo Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2gqv0gjpwzkyd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14618",
  "fulltext_url": null,
  "id": "2gqv0gjpwzkyd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14618",
  "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": "W7213251011",
  "pdf_url": "https://arxiv.org/pdf/2609.14618",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2gqv0gjpwzkyd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14618",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14618"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Quantitative Gidas--Ni--Nirenberg Theorem for the Fractional Laplacian",
  "title_text": "A Quantitative Gidas--Ni--Nirenberg Theorem for the Fractional Laplacian",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/2gqv0gjpwzkyd",
  "venue": "arXiv",
  "versions": []
}