{
  "abstract": "We establish a concentration--compactness theory for a one-dimensional singular Moser-type functional with logarithmic weight function $$\\sup\\limits_{v\\in E }\\int_0^1 e^{\\left(\\log\\frac{e}{s}+γ\\log\\log\\frac{e}{s}\\right)\\frac{v^2(s)}{s}}ds<\\infty,~γ\\in[0,1],$$ where $$E:= \\left\\{u \\in H^{1}(0, 1): u(0) = 0, \\int_0^1 |u'|^2dx=1\\right\\}.$$ In particular, we prove compactness for all normalized sequences in the subcritical range $γ<1$, thereby resolving an open problem proposed in [J. M. do Ó, B. Ruf and P. Ubilla, CVPDE 2023]. We also study the critical Bliss-Moser inequality: \\begin{equation}\\nonumber \\sup\\limits_{v\\in E }I^c(v)< +\\infty,\\quad I^c(v)=\\int_0^1 e^{\\left(\\log\\frac{e}{s}+\\log\\log\\frac{e}{s}+cs\\right)\\frac{v^2(s)}{s}}ds. \\end{equation} For any $c\\in(-\\infty,+\\infty)$, we determine the sharp concentration level $$\\sup_{v_j\\in E,v_j~\\text{is~NCS}}\\limsup\\limits_{j\\to\\infty} I^c(v_j)=1+e^2,$$ and obtain explicit sufficient conditions for attainability under $c>c^*$ and non-attainability under $c<-c_*$. This results describe how linear perturbations govern the transition between compactness, concentration, and attainability at the critical level.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23251",
  "author_total": 3,
  "authors": [
    {
      "name": "Yanyan Guo"
    },
    {
      "name": "Huxiao Luo"
    },
    {
      "name": "Bernhard Ruf"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4999he5bx7e9q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23251",
  "fulltext_url": null,
  "id": "4999he5bx7e9q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23251",
  "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": "W7214040853",
  "pdf_url": "https://arxiv.org/pdf/2609.23251",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4999he5bx7e9q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.23251",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23251"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Existence and non-existence of extremal functions for inequalities of Bliss-Moser type",
  "title_text": "Existence and non-existence of extremal functions for inequalities of Bliss-Moser type",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/4999he5bx7e9q",
  "venue": "arXiv",
  "versions": []
}