{
  "abstract": "In this paper we study an equivalence between the existence of a Sobolev inequality without gain, \\[\\|φ\\|_{L^p(v,Ω)} \\leq S(p,1) \\| \\sqrt{Q}\\nabla φ\\|_{L^p(Ω)},\\] that holds for smooth functions of compact support and the existence of a degenerate weak solution $(u,\\nabla u)\\in QH^{1,p}_0(v,Ω)$ to a Dirichlet problem for the $p$-Laplacian with a zero order term: \\begin{equation*} -v^{-1}\\text{Div}(|\\sqrt{Q}\\nabla u|^{p-2}Q\\nabla u)+F|u|^{p-2}u = |f|^{p-2}f - v^{-1}\\text{Div}(v|g|^{p-2}g\\mathbf{t}),\\; x \\in Ω, \\quad \\text{and} \\quad u =0, \\; x \\in \\partial Ω, \\end{equation*} More precisely, we use the Sobolev inequality to prove the existence of a degenerate weak solution to this equation and then use the existence of such a solution to produce a Sobolev inequality. Moreover, we show that solutions are unique.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22602",
  "author_total": 3,
  "authors": [
    {
      "name": "David Cruz-Uribe"
    },
    {
      "name": "Feyza Dal"
    },
    {
      "name": "Scott Rodney"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/782qzqx6m568h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22602",
  "fulltext_url": null,
  "id": "782qzqx6m568h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22602",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214011890",
  "pdf_url": "https://arxiv.org/pdf/2609.22602",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/782qzqx6m568h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22602",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22602"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sobolev Inequalities and the Existence of Solutions to Degenerate $p$-Poisson Equations",
  "title_text": "Sobolev Inequalities and the Existence of Solutions to Degenerate p-Poisson Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/782qzqx6m568h",
  "venue": "arXiv",
  "versions": []
}