{
  "abstract": "We consider positive weak solutions of $-Δ_p u=f(u)$, with $f$ positive and locally Lipschitz continuous. In the singular case $1<p<2$, the Harnack-type inequalities, the strong maximum principle for the linearized operator, the description of the critical set and the strong comparison principle established by Damascelli and Sciunzi [Calc.\\ Var.\\ Partial Differential Equations \\textbf{25} (2006), 139--159] have been available, for more than twenty years, only under the restriction $\\frac{2N+2}{N+2}<p<2$, a range that shrinks to the empty set as $N\\to\\infty$. We remove the dependence on the dimension and prove all of these results for $\\frac32<p<2$. Since the strong comparison principle is used as a black box throughout the qualitative theory of $p$-Laplace equations, the improvement propagates. We show in particular that the assumption $\\frac{2N+2}{N+2}<p<2$ may be replaced by $\\frac32<p<2$ in the resolution of Gibbons' conjecture for $-Δ_pu=f(u)$ by Esposito, Farina, Montoro and Sciunzi [Math.\\ Ann.\\ \\textbf{382} (2022), 943--974] and in their monotonicity theorem in half-spaces for changing-sign nonlinearities [Calc.\\ Var.\\ Partial Differential Equations \\textbf{61} (2022), art.\\ 154]; further applications are given in the body of the paper.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07877",
  "author_total": 1,
  "authors": [
    {
      "name": "Phuong Le"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/46jqg5p3r6ax3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07877",
  "fulltext_url": null,
  "id": "46jqg5p3r6ax3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07877",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7212025366",
  "pdf_url": "https://arxiv.org/pdf/2609.07877",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/46jqg5p3r6ax3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07877",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07877"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "An improved strong comparison principle for singular $p$-Laplace equations, with applications",
  "title_text": "An improved strong comparison principle for singular p-Laplace equations, with applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/46jqg5p3r6ax3",
  "venue": "arXiv",
  "versions": []
}