{
  "abstract": "Let $N \\geq 2$, $s\\in (0,1)$, and $p \\in (1, \\infty)$ with $N>sp$. For the fractional $p$-Laplacian, the power function $|x|^{-γ}$ with $0<γ<\\frac{N}{p-1}$ satisfies $$(-Δ_p)^s \\big(|x|^{-γ}\\big)=Φ(γ)|x|^{-γ(p-1)-sp} \\, \\text{ in }\\mathbb{R}^N\\setminus\\{0\\}.$$ Starting from the power function formula of Del Pezzo and Quaas (2026), we derive a paired integral representation of $Φ$, and show that $Φ$ is strictly increasing and then strictly decreasing, with a unique maximum at $\\frac{N-sp}{p}$. This maximum $\\barμ:=Φ(\\frac{N-sp}{p})$ equals one half of the sharp fractional Hardy constant. Consequently, for every $0<μ<\\barμ$, the equation $Φ(γ)=μ$ has exactly two roots $0<γ_{-}(μ)<\\frac{N-sp}{p}<γ_{+}(μ)<\\frac{N-sp}{p-1}$. These roots determine the homogeneous solutions $|x|^{-γ_{-}(μ)}$ and $|x|^{-γ_{+}(μ)}$ of the Hardy equation \\begin{equation*} (-Δ_p)^s u - μ\\frac{u^{p-1}}{|x|^{sp}}=0 \\quad \\text{in }\\mathbb R^N\\setminus\\{0\\}, \\; u> 0 \\text{ in } \\mathbb{R}^N \\setminus \\{0\\}. \\end{equation*} For $0<μ<\\barμ$, we then study the following critical Hardy-Sobolev equation \\begin{align}\\tag{$\\mathcal{P}$}\\label{P} (-Δ_p)^s u-μ\\frac{u^{p-1}}{|x|^{sp}} = u^{\\frac{Np}{N-sp}-1} \\text{ in } \\mathbb{R}^N, \\; u > 0 \\text{ in } \\mathbb{R}^N. \\end{align} Let $p \\geq 2$ and $U$ be a positive, radially symmetric, nonincreasing finite-energy weak solution to \\eqref{P}. We prove that there exists $C>0$ for which \\begin{align*} U(x) \\leq C|x|^{-γ_-(μ)}\\;\\text{as }|x| \\rightarrow 0,\\quad \\text{ and } \\quad U(x)\\le C|x|^{-γ_+(μ)} \\; \\text{as }|x| \\rightarrow \\infty. \\end{align*} Further, we show that $γ_-(μ)$ and $γ_+(μ)$ are the optimal exponents describing the singularity at the origin and the decay at infinity respectively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37212",
  "author_total": 2,
  "authors": [
    {
      "name": "Nirjan Biswas"
    },
    {
      "name": "Debangana Mukherjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7td0mz1v8kxfx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37212",
  "fulltext_url": null,
  "id": "7td0mz1v8kxfx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37212",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7224117375",
  "pdf_url": "https://arxiv.org/pdf/2610.37212",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7td0mz1v8kxfx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37212",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37212"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The homogeneity function of the fractional $p$-Laplacian and optimal asymptotic exponents for critical Hardy-Sobolev equations",
  "title_text": "The homogeneity function of the fractional p-Laplacian and optimal asymptotic exponents for critical Hardy-Sobolev equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/7td0mz1v8kxfx",
  "venue": "arXiv",
  "versions": []
}