{
  "abstract": "In this paper, we investigate the following quasi-linear weighted $N$-Laplacian Liouville equation \\begin{equation*}\\label{0} -Δ_N u=|x|^{Nα}e^{u}, \\qquad x\\in \\R^N, \\end{equation*} where $N \\geq 2$. For $\\al>0$, by carefully studying the linearized problem and applying the approximation method and bifurcation theory, we prove that, when the parameter $α$ equals to the critical values $α(k):=\\frac{\\sqrt{k(N-1)(k+N-2)}}{N-1}-1$ for $k \\geq 2$, there exist non-radial solutions $u$ (bifurcating from $U_{α(k)}$) to the above quasi-linear Hénon type Liouville equation such that $u\\sim \\ln|x|$, $|\\nabla u|= O(|x|^{-1})$ at $\\infty$ and $\\int_{\\R^N}|x|^{Nα}e^{u}\\md x=N\\left(\\frac{N^2}{N-1}\\right)^{N-1}(α+1)^{N-1}ω_N$. One should note that, $α(k)=k-1$ for $k\\geq2$ when $N=2$. Our results successfully extend the existence result of J. Prajapat and G. Tarantello in \\cite{PT} concerning the $2$-dimension and Laplacian case (i.e., $N=2$) to the more general $N$-dimension and $N$-Laplacian cases ($N\\geq 2$), and extend the results of F. Gladiali, M. Grossi, and S. L. N. Neves in \\cite{GGN} and the authors in \\cite{DDGL} from $1<p<N$ to the much more complicated limiting case $p=N$. We introduced some new ideas and overcame a series of crucial difficulties, including the nonlinearity nature of the $N$-Laplacian $Δ_N$, the lack of Green integral representation formula and critical weighted Sobolev embedding inequality, the absence of Kelvin type transforms for linearized/difference equations, the invariance of the total mass under scalings of $u$, and the signs-changing and divergence (to $-\\infty$) at $\\infty$ of the solutions, which makes the suitable choices of the approximate problems, the (normalized) approximate function sequences and the working space to be quite difficult.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.24012",
  "author_total": 4,
  "authors": [
    {
      "name": "Wei Dai"
    },
    {
      "name": "Lixiu Duan"
    },
    {
      "name": "Changfeng Gui"
    },
    {
      "name": "Yuan Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cwj4mhbgnw113.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.24012",
  "fulltext_url": null,
  "id": "cwj4mhbgnw113",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.24012",
  "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": "W7171572919",
  "pdf_url": "https://arxiv.org/pdf/2607.24012",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cwj4mhbgnw113.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2607.24012",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.24012"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Non-radial solutions for the quasi-linear Hénon type $N$-Laplacian Liouville equation",
  "title_text": "Non-radial solutions for the quasi-linear Hénon type N-Laplacian Liouville equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/cwj4mhbgnw113",
  "venue": "arXiv",
  "versions": []
}