{
  "abstract": "In this paper, we study the critical Moser--Trudinger supremum \\[ S(λ,p) = \\sup_{\\substack{ u\\in H^1(\\mathbb{R}^2)\\\\ \\int_{\\mathbb{R}^2} (|\\nabla u|^2+u^2)\\,dx\\le 1 }} \\int_{\\mathbb{R}^2} \\left(e^{4πu^2}-1-λ|u|^p\\right)\\,dx, \\qquad p\\ge 2. \\] Building on the classification away from the thresholds by Chen, Jiang, Lu and Zhu (2026), we resolve the endpoint cases: both thresholds for $p=2$ and the finite threshold for $2<p\\le4$ are attained. We also prove compactness of centred radial extremals and one-sided value expansions governed by the selected endpoint $L^p$ mass. For $p>4$, we quantify how extremals spread out as $λ\\to\\infty$ by establishing the two-term expansion \\[ S(λ,p)=4π+\\mathcal C_pλ^{-2/(p-4)} +D_pλ^{-4/(p-4)}+o\\bigl(λ^{-4/(p-4)}\\bigr), \\] Here $\\mathcal C_p$ is the maximum of a limiting variational problem at unit $L^2$ mass. An explicit correction involving the sixth moment and the full $H^1$ normalization determines $D_p$ and selects the limiting maximizers. Rescaled centred radial extremals converge strongly to these profiles in $H^1(\\mathbb{R}^2)$ along subsequences. For every fixed $λ>λ_*(4)$, we further establish \\[ \\lim_{p\\downarrow4}(p-4)\\log\\bigl(S(λ,p)-4π\\bigr) =2\\log\\frac{λ_{\\mathrm{loc}}}λ<0, \\] where $λ_{\\mathrm{loc}}$ is the local critical penalty determined by the sharp Gagliardo--Nirenberg constant. The gradient energy has the same logarithmic rate, and the centred radial profiles, normalized in mass and gradient energy, converge along subsequences to sharp Gagliardo--Nirenberg optimizers. An exact transfer identity extends these endpoint and asymptotic results to every constant planar norm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29402",
  "author_total": 2,
  "authors": [
    {
      "name": "Kaiwen Guo"
    },
    {
      "name": "Yanjun Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fvsp1beq7jxew.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29402",
  "fulltext_url": null,
  "id": "fvsp1beq7jxew",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29402",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7223228398",
  "pdf_url": "https://arxiv.org/pdf/2610.29402",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fvsp1beq7jxew.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29402",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29402"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Threshold Attainment and Sharp Asymptotics for $L^p$-Perturbed Moser--Trudinger Functionals on $\\mathbb{R}^2$",
  "title_text": "Threshold Attainment and Sharp Asymptotics for L^p-Perturbed Moser--Trudinger Functionals on ℝ^2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/fvsp1beq7jxew",
  "venue": "arXiv",
  "versions": []
}