{
  "abstract": "In this paper, we study local and global properties of positive solutions to the equation $-Δu=u^p|\\nabla u|^q$ in a domain $Ω$ of $\\mathbb R^N$, where $p$ and $q$ are parameters. We introduce a linear operator to construct the differential inequality to obtain gradient estimates, and further establish Liouville-type theorems. As an application, we derive universal estimates for local solutions. Some of our results are new, as we extend the condition $p+q<(N+3)/(N-1)$ considered by He, Hu and Wang [Math. Z. 313 (2026), No. 6] to a wider range of parameters.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17149",
  "author_total": 2,
  "authors": [
    {
      "name": "Wenguo Liang"
    },
    {
      "name": "Zhengce Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/851nezm05kvhj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17149",
  "fulltext_url": null,
  "id": "851nezm05kvhj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17149",
  "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": "W7213412919",
  "pdf_url": "https://arxiv.org/pdf/2609.17149",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/851nezm05kvhj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17149",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17149"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient",
  "title_text": "Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/851nezm05kvhj",
  "venue": "arXiv",
  "versions": []
}