{
  "abstract": "We study minimizers of the optimal Sobolev quotients for the critical embeddings $D^{1,p}(\\mathbb R^N;\\mathbb R^m)\\hookrightarrow L^{p^*}(\\mathbb R^N;\\mathbb R^m)$ and $W^{1,p}(Ω;\\mathbb R^m)\\hookrightarrow L^{p^*}(Ω;\\mathbb R^m)$. In the whole space, we classify all minimizers as $\\boldsymbol u=\\boldsymbol c\\,U_{δ,x_0}$, where $\\boldsymbol c\\neq\\boldsymbol0$ and $U_{δ,x_0}$ is an Aubin--Talenti bubble. On bounded domains and for $λ>0$, we equip $W^{1,p}(Ω;\\mathbb R^m)$ with $ \\|\\boldsymbol u\\|_λ^p :=\\int_Ω \\bigl(|D\\boldsymbol u|^p+λ|\\boldsymbol u|^p\\bigr) $. Using scalar attainment results, we establish existence and synchronization of vectorial minimizers. At resonance, we consider the pure Neumann problem $ -\\boldsymbolΔ_p\\boldsymbol u =|\\boldsymbol u|^{p^*-2}\\boldsymbol u\\ \\text{in }Ω. $ For the associated quotient, we prove existence and global $C^1(\\overlineΩ)$ regularity of minimizers. Finally, we prove synchronization of radial solutions and deduce symmetry breaking on balls.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18002",
  "author_total": 1,
  "authors": [
    {
      "name": "Simone Mauro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2y7meznssjdnz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18002",
  "fulltext_url": null,
  "id": "2y7meznssjdnz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18002",
  "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": "W7221786710",
  "pdf_url": "https://arxiv.org/pdf/2610.18002",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2y7meznssjdnz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.18002",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18002"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sobolev Minimizers for Critical Vectorial $p$-Laplacian Problems",
  "title_text": "Sobolev Minimizers for Critical Vectorial p-Laplacian Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/2y7meznssjdnz",
  "venue": "arXiv",
  "versions": []
}