{
  "abstract": "We work on connected, noncompact $n$-dimensional Riemannian manifolds $(\\mathcal{M}, g), n \\geq 3$, that are complete with respect to $d_g$ and have noncompact boundary and bounded geometry. We study the existence of positive solutions in $C^{\\infty}(\\mathcal{M}) \\cap L^{2^*}(\\mathcal{M}) \\cap L^{\\infty}(\\mathcal{M})$ to the Yamabe problem: $$ \\left\\{\\begin{aligned} -Δ_g u+\\frac{n-2}{4(n-1)} R_g u & =\\mathscr{C}_0 u^{\\frac{n+2}{n-2}} \\quad \\text { in } \\mathcal{M}, \\\\ \\frac{\\partial u}{\\partial ν}+\\frac{n-2}{2(n-1)} H_g u & =\\mathscr{C}_1 u^{\\frac{n}{n-2}} \\quad \\text { on } \\partial \\mathcal{M}, \\end{aligned}\\right. $$ where $\\mathscr{C}_0=Y_{\\mathscr{C}}(\\mathcal{M}), \\mathscr{C}_1=Y_{\\mathscr{C}}(\\mathcal{M}) \\mathscr{C} \\bar{d}^{\\frac{2}{n-2}}$, the parameter $\\mathscr{C} \\geq 0$ is fixed, and $$ Y_{\\mathscr{C}}(\\mathcal{M}):=\\inf _{u \\in C_c^{\\infty}(\\mathcal{M}) \\backslash\\{0\\}} \\frac{E_g(u)}{\\left(\\|u\\|_{L^{2^*}(\\mathcal{M})}^{2^*}+\\mathscr{C}\\|u\\|_{L^{2^{\\#}}(\\partial \\mathcal{M})}^{2^*}\\right)^{2 / 2^*}} . $$ Here $E_g$ is the associated quadratic energy. We assume $Y_{\\mathscr{C}}(\\mathcal{M})<Y_{\\mathscr{C}, \\infty}(\\mathcal{M})$ and $Y_{\\mathscr{C}, \\infty}(\\mathcal{M})>0$, where $Y_{\\mathscr{C}, \\infty}$ is the corresponding variational level at infinity. We approach this problem using nonnegative solutions $u_j \\in W_{\\partial}^{1,2}\\left(B_g(\\mathcal{O}, j)\\right), j \\in \\mathbb{N}, j \\geq j_0$, for which we prove that $$ \\sup _{j \\geq j_0}\\left\\|u_j\\right\\|_{L^{\\infty}\\left(B_g(\\mathcal{O}, j-1)\\right)}<\\infty . $$ For $\\mathscr{C}>0$, we also prove that, along a subsequence, $\\left\\|u_j\\right\\|_{L^2{ }_{\\left(B_g(\\mathcal{O}, j) \\cap \\partial \\mathcal{M}\\right)}} \\rightarrow \\bar{d}$, with $0<\\bar{d} \\leq \\mathscr{C}^{-1 / 2^*}$. For $\\mathscr{C}=0$, we set $\\bar{d}=0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40603",
  "author_total": 1,
  "authors": [
    {
      "name": "Juan Pablo Alcon Apaza"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0rbfpdgcqn6b8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40603",
  "fulltext_url": null,
  "id": "0rbfpdgcqn6b8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40603",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40603",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0rbfpdgcqn6b8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:35Z",
      "native_id": "2610.40603",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40603"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Critical Robin Problems on Manifolds with Noncompact Boundary",
  "title_text": "Critical Robin Problems on Manifolds with Noncompact Boundary",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:53Z",
  "url": "https://research.draconitic.com/paper/0rbfpdgcqn6b8",
  "venue": "arXiv",
  "versions": []
}