{
  "abstract": "This paper establishes a new comparison principle framework for the subquadratic fractional $p$-Laplacian, i.e.~$1 < p < 2$ under minimal regularity assumptions, that has remained a significant challenging issue due to the singularity of the operator. Our results provide the essential analytical tools required for the moving plane method in this setting. We prove first a weak comparison principle for $(-Δ)_p^s u = f(u)$ in bounded domains with sufficiently small measure, where only the boundedness of the weak solution is required. More importantly, we establish a strong comparison principle for continuous weak solutions in the parameter range $s \\in (0, \\frac{1}{2})$ and $\\frac{1}{1-s} < p < 2$. Our proof introduces a localized barrier function and does not require any Hölder regularity of the weak solution, nor any smoothness of the domain. This presents a substantial contrast over previous study, which relied heavily on Hölder or even $C^{1,1}$ regularity. As a direct application, we employ these comparison principles to prove the symmetry of weak solutions to $(-Δ)_p^s u = f(u)$ under mild assumptions, which significantly extend existing symmetry theories for nonlocal quasilinear equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02626",
  "author_total": 2,
  "authors": [
    {
      "name": "Dong Ye"
    },
    {
      "name": "Weimin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c77fpq8aps117.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02626",
  "fulltext_url": null,
  "id": "c77fpq8aps117",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02626",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7207827537",
  "pdf_url": "https://arxiv.org/pdf/2609.02626",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c77fpq8aps117.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02626",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02626"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Comparison principles and symmetry for subquadratic fractional $p$-Laplacian equations",
  "title_text": "Comparison principles and symmetry for subquadratic fractional p-Laplacian equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/c77fpq8aps117",
  "venue": "arXiv",
  "versions": []
}