{
  "abstract": "In this article, we investigate the following modified quasilinear equation with parameter driven by the $Q$-subLaplacian: \\begin{align*} \\begin{cases} -Δ_Q u - Δ_Q\\bigl(|u|^{2α}\\bigr)\\,|u|^{2α-2} u = λf(ξ,u) & \\text{in } Ω, \\\\[2mm] u = 0 & \\text{on } \\partialΩ, \\end{cases} \\end{align*} where $Δ_Q(\\cdot):= \\mathrm{div}_{\\mathbb{H}}\\bigl(|\\nabla_{\\mathbb{H}}(\\cdot)|^{Q-2}\\nabla_{\\mathbb{H}}(\\cdot)\\bigr)$ denotes the $Q$-subLaplacian on the Heisenberg group $\\mathbb{H}^N$, $Q=2N+2$ is the homogeneous dimension, $Ω\\subset \\mathbb{H}^N$ is a smooth bounded domain, $λ>0$, $α> \\frac{1}{2}$, and $f$ has critical or subcritical exponential growth of order $\\exp\\bigl(β|t|^{2αQ/(Q-1)}\\bigr)$. We prove three results: the existence of a nontrivial positive weak solution in the critical case for all large $λ$, and the existence of a least-energy nodal solution with exactly two nodal domains, under subcritical and critical exponential growth. A change of variables $u=g(v)$ reduces the problem to a quasilinear problem whose energy functional is of class $C^1$; the exponent $2αQ/(Q-1)$ arises from the growth of $g$. We handled the exponential growth using the sharp Moser-Trudinger inequality of Cohn and Lu. The positive solution is obtained by the mountain pass theorem and the nodal solutions by minimization on a nodal Nehari set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.14804",
  "author_total": 3,
  "authors": [
    {
      "name": "Ankit Mishra"
    },
    {
      "name": "Sarika Goyal"
    },
    {
      "name": "Divya Goel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4jq2xxs2v2nh4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.14804",
  "fulltext_url": null,
  "id": "4jq2xxs2v2nh4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.14804",
  "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": "W7169504985",
  "pdf_url": "https://arxiv.org/pdf/2607.14804",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4jq2xxs2v2nh4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2607.14804",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.14804"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Positive and nodal solutions for a parametric quasilinear $Q$-sub-Laplacian problem with critical exponential growth on the Heisenberg group",
  "title_text": "Positive and nodal solutions for a parametric quasilinear Q-sub-Laplacian problem with critical exponential growth on the Heisenberg group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/4jq2xxs2v2nh4",
  "venue": "arXiv",
  "versions": []
}