{
  "abstract": "For $1<p<2$ we prove weak and strong unique continuation properties for the solutions to $Δ_pu+f(u)=0$ in a planar domain, with $f$ continuous and such that $|f(s)|\\leq C|s|^{p-1}$: a solution vanishing on an open set vanishes identically, and so does a solution vanishing to infinite order at a single point. We can therefore make progress towards a proof of some long-standing nonexistence results available only for $p=2$, such as a celebrated one of Esteban and Lions, establishing here that for the critical $p$-Laplace equation with zero Dirichlet boundary condition on a half-plane, there are no nontrivial finite energy solutions. In the plane, unique continuation thus provides the missing ingredient for a generalisation to the $p$-Laplacian operator of a classical result of Struwe on the bubble-profile decomposition of possibly sign-changing Palais-Smale sequences associated to the Brezis-Nirenberg problem, which remains open in dimension $N\\geq3$ for $p\\neq2$. We obtain a characterisation of their loss of compactness in terms of the finite energy solutions of $Δ_pu+|u|^{p^*-2}u=0$ in $\\mathbb R^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34204",
  "author_total": 1,
  "authors": [
    {
      "name": "Carlo Mercuri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7sc5e7ndhm8kj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34204",
  "fulltext_url": null,
  "id": "7sc5e7ndhm8kj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34204",
  "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": "W7214757837",
  "pdf_url": "https://arxiv.org/pdf/2609.34204",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7sc5e7ndhm8kj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.34204",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34204"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Unique continuation, nonexistence and bubbling for the critical $p$-Laplace equation in the plane",
  "title_text": "Unique continuation, nonexistence and bubbling for the critical p-Laplace equation in the plane",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/7sc5e7ndhm8kj",
  "venue": "arXiv",
  "versions": []
}