{
  "abstract": "Let $ C_ω$ be the cone generated by a proper $C^2$ domain $ω\\subset\\mathbb S^{N-1}$. We study positive homogeneous solutions of the fractional $p$-Laplace equation in $C_ω$ with zero exterior values. For $p\\ne2$, the operator is nonlinear and not uniformly elliptic. For every $p>1$, we identify two extremal homogeneities $β^+>0>β^-$, prove that every fundamental solution, if one exists, has one of these exponents, and construct homogeneous eigenfunctions whose eigenvalues have signs determined by the position of the homogeneity relative to $β^\\pm$. As an application, we obtain the Lane-Emden threshold $q^*=p-1+sp/β^+$ for every $p>1$: nontrivial nonnegative supersolutions of $(-Δ_p)^s u\\ge u^q$ in $C_ω$ do not exist for $0<q<q^*$, whereas homogeneous supersolutions exist for $q>q^*$. We also establish Hopf-type lower bounds at conical boundary points with the optimal exponent. When $p-1>sp$ , we construct both fundamental solutions, one singular at the vertex and the other vanishing there. For $p=2$, these solutions are known; our method recovers them for every $s\\in(0,1)$. The homogeneous eigenfunctions also serve as barriers, yielding boundary decay and global Hölder regularity for $(-Δ_p)^s u=f$ in bounded Lipschitz domains, with zero exterior values and $f$ continuous up to the boundary. The admissible Hölder exponents are determined by the exterior cones. All results hold for every $N\\ge2$; in particular, $N>sp$ is not needed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07157",
  "author_total": 2,
  "authors": [
    {
      "name": "Reshmi Biswas"
    },
    {
      "name": "Alexander Quaas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5x4n8wynxj0sr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07157",
  "fulltext_url": null,
  "id": "5x4n8wynxj0sr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07157",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07157",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5x4n8wynxj0sr.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07157",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07157"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Fundamental solutions of the fractional $p$-Laplacian in cone-like domains, with applications to Liouville-type theorems and global Hölder regularity",
  "title_text": "Fundamental solutions of the fractional p-Laplacian in cone-like domains, with applications to Liouville-type theorems and global Hölder regularity",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/5x4n8wynxj0sr",
  "venue": "arXiv",
  "versions": []
}