{
  "abstract": "Let $Ω\\subset\\mathbb R^2$ be a smooth bounded domain containing the origin and invariant under reflection across the coordinate axes, and let $0<λ<λ_1(Ω)$, where $Λ_1 (Ω)$ is the first eigenvalue for $-Δ$ on $Ω$ under Dirichlet boundary conditions. For every fixed integer $k\\ge1$ and all sufficiently small $\\varepsilon>0$, we construct a positive solution of \\[ -Δu=λu e^{u^{2+\\varepsilon}} \\quad\\hbox{in }Ω, \\qquad u=0\\quad\\hbox{on }\\partialΩ, \\] which blows up at the origin as a tower of $k$ Liouville bubbles. The concentration scales are strongly separated. The proof is based on a Lyapunov--Schmidt reduction adapted to these different scales.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22413",
  "author_total": 3,
  "authors": [
    {
      "name": "Manuel del Pino"
    },
    {
      "name": "Ignacio Guerra"
    },
    {
      "name": "Monica Musso"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/34k0x5hf63bt4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22413",
  "fulltext_url": null,
  "id": "34k0x5hf63bt4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22413",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213997834",
  "pdf_url": "https://arxiv.org/pdf/2609.22413",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/34k0x5hf63bt4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22413",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22413"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Supercritical Problem in Dimension Two",
  "title_text": "A Supercritical Problem in Dimension Two",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/34k0x5hf63bt4",
  "venue": "arXiv",
  "versions": []
}