{
  "abstract": "We study the existence and qualitative properties of non-negative solutions to singular 1-Laplacian problems of the form \\begin{equation*} \\begin{cases} -Δ_1 u = g(u)|D u| + h(u)f & \\text{in } Ω,\\newline u=0 & \\text{on } \\partialΩ, \\end{cases} \\end{equation*} where $Ω\\subset \\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $0<f\\in L^{N,\\infty}(Ω)$, and $g,h$ are continuous positive functions that may blow up at the origin. We establish the existence of a positive bounded solution even when $g$ is non-integrable at zero and $h(0)=+\\infty$, breaking the non-existence phenomenon appearing in the associated $p$-Laplacian problems. Furthermore, we prove that solutions possess a non-zero boundary trace on a set of positive measure of $\\partialΩ$ whenever $g(s)$ grows sufficiently fast near $s=0$; by contrast, when this growth condition fails, solutions vanishing on the boundary may appear. Finally, when $h(0)<+\\infty$, we provide a sharp threshold classification of trivial and non-trivial solutions depending on the size of $f$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22027",
  "author_total": 3,
  "authors": [
    {
      "name": "Antonio J. Martínez Aparicio"
    },
    {
      "name": "Francescantonio Oliva"
    },
    {
      "name": "Francesco Petitta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/11thyg8548qc4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22027",
  "fulltext_url": null,
  "id": "11thyg8548qc4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22027",
  "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": "W7222702141",
  "pdf_url": "https://arxiv.org/pdf/2610.22027",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/11thyg8548qc4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22027",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22027"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Existence and classification of solutions for singular 1-Laplacian equations with gradient terms",
  "title_text": "Existence and classification of solutions for singular 1-Laplacian equations with gradient terms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/11thyg8548qc4",
  "venue": "arXiv",
  "versions": []
}