{
  "abstract": "We study the singular limit $p \\downarrow 1$ for positive radial entire solutions $u_p$ of $$ -Δ_p u_p=F_p\\left(|x|, u_p,\\left|\\nabla u_p\\right|\\right) u_p^{-β_p} \\quad \\text { in } \\mathbb{R}^n, $$ where $n \\geq 2,1<p<\\min \\{2, \\sqrt{n}\\}$, and $0 \\leq β_p \\leq p-1$. Assuming that $$ F_p \\rightarrow F_1 \\quad \\text { locally uniformly }, \\quad \\frac{β_p}{p-1} \\rightarrow \\mathbf{c} \\in[0,1] \\quad \\text { as } p \\downarrow 1, $$ we prove that, after passing to a subsequence, $u_{p_j} \\rightarrow u$ in $C_{\\text {loc }}\\left(\\mathbb{R}^n\\right)$ and $\\nabla u_{p_j} \\stackrel{*}{\\rightharpoonup} \\nabla u$ in $L_{\\text {loc }}^{\\infty}\\left(\\mathbb{R}^n ; \\mathbb{R}^n\\right)$, where $u \\in W^{1, \\infty}\\left(\\mathbb{R}^n\\right)$ is radial, and solves $$ -Δ_1 u=\\mathtt{K}(|x|) \\quad \\text { in } \\mathbb{R}^n . $$ On $\\left\\{x \\in \\mathbb{R}^n \\mid \\mathtt{G}(|x|)<1\\right\\}$, we prove that $\\mathtt{K}(|x|)=w^{-\\mathbf{c}} F_1(|x|, u(x), 0)$ and that $\\nabla u=0$ a.e., where $u_{p_i}^{p_i-1} \\rightarrow w$. On the other hand, we show that if $ρ\\mapsto F_1(r, t, ρ)$ is affine for every $r$ and $t$, then $\\mathtt{K}(|x|)=$ $w^{-\\mathbf{c}} F_1(|x|, u(x),|\\nabla u(x)|)$ a.e. in $\\mathbb{R}^n$. For $n=1$ and $1<p<2$, we study $$ \\frac{\\mathrm{d}}{\\mathrm{~d} x}\\left(\\left|u_p^{\\prime}\\right|^{p-2} u_p^{\\prime}\\right)=\\mathtt{F}_p\\left(|x|, u_p,\\left|u_p^{\\prime}\\right|\\right) u_p^{-β_p} \\quad \\text { in } \\mathbb{R} $$ and obtain analogous results as $p \\downarrow 1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17511",
  "author_total": 1,
  "authors": [
    {
      "name": "Juan Pablo Alcon Apaza"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7bm8y0t3fdqx6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17511",
  "fulltext_url": null,
  "id": "7bm8y0t3fdqx6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17511",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213436179",
  "pdf_url": "https://arxiv.org/pdf/2609.17511",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7bm8y0t3fdqx6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17511",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17511"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Asymptotic Behavior of Radial Solutions to Singular $p$-Laplacian Equations as $p\\to1$",
  "title_text": "Asymptotic Behavior of Radial Solutions to Singular p-Laplacian Equations as p→1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/7bm8y0t3fdqx6",
  "venue": "arXiv",
  "versions": []
}