{
  "abstract": "Let $1<m<n$ and $0<q<m-1$. We classify positive, locally bounded weak solutions of \\[-Δ_m u=u^p|Du|^q\\quad\\text{in }\\mathbb R^n,\\qquad p=\\frac{m-q}{n-m}\\left(n+\\frac{q}{m-1-q}\\right)-1.\\] Every solution is constant or belongs to an explicit family of radial solutions, up to translation and scaling. No global bound, decay, or finite-energy assumption is required. The proof is based on a scalar differential identity and a maximum principle that remains valid at critical points. These lead to a sharp gradient bound, classification of normalized solutions, and a first-contact argument for arbitrary entire solutions. Local regularity and the behavior at critical points are treated throughout in the weak solution class; global $C^2$ regularity is not assumed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33583",
  "author_total": 3,
  "authors": [
    {
      "name": "Tian Wu"
    },
    {
      "name": "Jin Yan"
    },
    {
      "name": "Hua Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bfhs142sjse56.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33583",
  "fulltext_url": null,
  "id": "bfhs142sjse56",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33583",
  "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": "W7214713289",
  "pdf_url": "https://arxiv.org/pdf/2609.33583",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bfhs142sjse56.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33583",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33583"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The $m$-Laplace equation with a gradient term II: classification in the second critical case",
  "title_text": "The m-Laplace equation with a gradient term II: classification in the second critical case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/bfhs142sjse56",
  "venue": "arXiv",
  "versions": []
}