{
  "abstract": "We study the following degenerate elliptic equation with combined nonlinearities and prescribed mass \\begin{equation*} \\begin{aligned} \\begin{cases} -Δ_γu+ωu+|u|^{p-2}u=|u|^{q-2}u \\ \\ \\mbox{in }\\mathbb{R}^{N}, \\\\ \\int_{\\mathbb{R}^{N}} |u|^{2} \\mathrm{d}z=c, \\end{cases} \\end{aligned} \\end{equation*} where $Δ_γ$ is the Grushin operator and $ω\\in\\mathbb{R}$ appears as an unknown Lagrange multiplier. We characterize the critical mass threshold and the energy ground state solutions. Firstly, we prove the existence of optimizers for the Gagliardo-Nirenberg inequality involving the Grushin operator together with estimates for the related best constant. Secondly, we characterize the critical mass by the best constant of Gagliardo-Nirenberg inequality and prove the existence of $x$-radial energy ground state solutions. Finally, we describe the relation between the $x$-radial action and energy ground state solutions, and the optimizers of Gagliardo-Nirenberg inequality. The strategy developed here is based on the interplay between the mass-preserving scaling and a sharp Gagliardo-Nirenberg inequality. It therefore suggests possible extensions to other prescribed-mass problems with competing homogeneous terms and threshold-mass phenomena.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36927",
  "author_total": 3,
  "authors": [
    {
      "name": "Aleks Jevnikar"
    },
    {
      "name": "Senli Liu"
    },
    {
      "name": "Yu Su"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7z6z5xw1h0adh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36927",
  "fulltext_url": null,
  "id": "7z6z5xw1h0adh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36927",
  "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": "W7224077213",
  "pdf_url": "https://arxiv.org/pdf/2610.36927",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7z6z5xw1h0adh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36927",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36927"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A class of degenerate elliptic problems involving the Grushin operator: critical mass and ground states",
  "title_text": "A class of degenerate elliptic problems involving the Grushin operator: critical mass and ground states",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/7z6z5xw1h0adh",
  "venue": "arXiv",
  "versions": []
}