{
  "abstract": "In this paper, we investigate the singular Dirichlet problem \\begin{equation}\\label{abs1} \\left\\{ \\begin{array}{rclr} -Δ_1u &=&\\dfracλ{|x|}\\hbox{Sgn}\\,(u)+\\dfrac{f}{u^γ}&\\quad\\mbox{in}\\; Ω,\\\\[1ex] u &=&0& \\quad\\mbox{on}\\; \\partialΩ, \\end{array} \\right. \\end{equation} where $Δ_1u=\\hbox{div}\\left(\\frac{Du}{|Du|}\\right)$ denotes the $1$-Laplacian operator, the parameters are $λ\\in {R}$ and $γ>0$, and $f$ is a non-negative function belonging to the Lorentz space $L^{N,\\infty}(Ω)$. Our main goal is to establish the existence of non-trivial solutions under the sharp restriction $λ<N-1$, regardless of the magnitude of the datum $f$ or the value of the singular exponent $γ>0$. We also show that these solutions are globally bounded and that, on the contrary, no solution exists as soon as $λ\\geq N-1$ and $f$ is positive. These results are achieved by rigorously analyzing the asymptotic behavior, as $p\\to 1^+$, of the solutions to the approximating $p$-Laplace problems \\begin{equation} \\left\\{ \\begin{array}{rclc} -Δ_pu&=&\\dfracλ{|x|^p}|u|^{p-2}u+\\dfrac{f}{u^γ}&\\quad\\mbox{ in }\\; Ω,\\\\[1ex] u&=&0&\\quad\\mbox{ on }\\; \\partialΩ. \\end{array} \\right. \\end{equation} Finally, we provide a family of explicit examples designed to illustrate the sharp optimality of our main assumptions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28298",
  "author_total": 2,
  "authors": [
    {
      "name": "Juan Carlos Chata Ortiz"
    },
    {
      "name": "Francesco Petitta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/61s31e54qaak5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28298",
  "fulltext_url": null,
  "id": "61s31e54qaak5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28298",
  "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": "W7214148866",
  "pdf_url": "https://arxiv.org/pdf/2609.28298",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/61s31e54qaak5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28298",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28298"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp existence and non-existence for singular $1$-Laplace equations with Hardy potentials",
  "title_text": "Sharp existence and non-existence for singular $1$-Laplace equations with Hardy potentials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/61s31e54qaak5",
  "venue": "arXiv",
  "versions": []
}