{
  "abstract": "We study positive solutions of $Iu+μu=k(x)u^p$ in a bounded domain, where $p>1$, the nonnegative coefficient $k$ may vanish on a sanctuary region, and $I$ is a fully nonlinear nonlocal operator uniformly elliptic in the Caffarelli--Silvestre sense. We characterize the sharp existence interval in terms of positive principal half-eigenvalues and prove uniqueness. At the lower endpoint, the solutions vanish uniformly and their normalized profiles converge to the principal eigenfunction. If $k$ vanishes on a sanctuary $Ω_0$, then, as $μ\\uparrowλ(Ω_0)$, the solutions blow up locally uniformly in $Ω$. Writing $ϕ_0=ϕ(\\cdot,Ω_0)$ for the normalized positive principal eigenfunction in $Ω_0$, we prove \\[ \\frac{u_μ}{\\|u_μ\\|_{L^\\infty(Ω)}}\\toϕ_0, \\qquad \\frac{u_μ}{\\|u_μ\\|_{L^\\infty(Ω)}^{1/p}} \\to\\left(\\frac{Iϕ_0}{k}\\right)^{1/p}, \\] uniformly in $\\mathbb{R}^n$ and locally uniformly in $Ω\\setminus\\overline{Ω_0}$, respectively. Thus the upper-endpoint blow-up occurs on two distinct scales.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09271",
  "author_total": 2,
  "authors": [
    {
      "name": "Giulio Galise"
    },
    {
      "name": "Shuhei Kitano"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a3sghegyxeped.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09271",
  "fulltext_url": null,
  "id": "a3sghegyxeped",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09271",
  "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": "W7221041101",
  "pdf_url": "https://arxiv.org/pdf/2610.09271",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a3sghegyxeped.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09271",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09271"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Fully Nonlinear Logistic Equations with Sanctuary Regions and Nonlocal Diffusion",
  "title_text": "Fully Nonlinear Logistic Equations with Sanctuary Regions and Nonlocal Diffusion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/a3sghegyxeped",
  "venue": "arXiv",
  "versions": []
}