{
  "abstract": "We study a singular Dirichlet problem driven by a regularized $p(x)$-Laplacian under a prescribed modular constraint. Using growth and differentiability estimates for the regularized energy, together with coercivity and modular compactness, we construct nonnegative constrained minimizers for a family of nonsingular approximating functionals. Under the admissibility of signed constrained variations, each minimizer satisfies an approximate Euler--Lagrange equation with a uniquely determined constraint multiplier. An explicit uniform boundary lower barrier and a compatible weight condition, combined with a variable exponent Hardy inequality, provide global dual control of the singular sources and convergence against every zero-trace Sobolev test function. For these approximate minimizing pairs, the singular-limit result gives subsequential strong Sobolev convergence of the states and convergence of their multipliers, while preserving positivity and the prescribed modular. The limiting pair is therefore a positive normalized weak solution of the singular problem, obtained by removing the reaction regularization while keeping the gradient regularization fixed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20778",
  "author_total": 1,
  "authors": [
    {
      "name": "Mustafa Avci"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bqh5aw4s3p48m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20778",
  "fulltext_url": null,
  "id": "bqh5aw4s3p48m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20778",
  "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": "W7213582454",
  "pdf_url": "https://arxiv.org/pdf/2609.20778",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bqh5aw4s3p48m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20778",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20778"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Positive normalized solutions for a singular regularized $p(x)$-Laplacian Dirichlet problem",
  "title_text": "Positive normalized solutions for a singular regularized p(x)-Laplacian Dirichlet problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/bqh5aw4s3p48m",
  "venue": "arXiv",
  "versions": []
}