{
  "abstract": "We consider the Lane-Emden problem $-Δu=|u|^{p-1}u$, $p>1$, in bounded domains of a half-cylinder $ω\\times(0,+\\infty)$, with Dirichlet conditions on the relative boundary and Neumann conditions on the boundary of the cylinder. In flat cylinders $ω\\times(0,L)$ this problem admits, for every $p>1$, one-dimensional solutions with any prescribed number $m$ of nodal regions. We study the stability of these domains, as critical points of a related energy functional, with respect to volume-preserving deformations of the domain. We prove that, for $p$ large, the flat cylinders are stable as soon as their height exceeds a threshold growing linearly in $p$, while, for $p$ close to $1$, both stability and instability occur, with explicit thresholds depending on $m$ and on the first nontrivial Neumann eigenvalue of $ω$. These results rely on a detailed analysis of the one-dimensional profiles: we describe their asymptotic behaviour, and that of the first eigenvalue of the associated linearized operator, as $p\\to1^+$ and $p\\to+\\infty$, we obtain expansions of the amplitude with explicit constants, and we prove global comparison results with respect to $p$, some of which are new also for positive solutions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06339",
  "author_total": 2,
  "authors": [
    {
      "name": "Francesca De Marchis"
    },
    {
      "name": "Lisa Mazzuoli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7wgnecrs64j1n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06339",
  "fulltext_url": null,
  "id": "7wgnecrs64j1n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06339",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06339",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7wgnecrs64j1n.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06339",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06339"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "One-dimensional nodal solutions of the Lane--Emden problem in cylinders: asymptotic behaviour and energy stability",
  "title_text": "One-dimensional nodal solutions of the Lane--Emden problem in cylinders: asymptotic behaviour and energy stability",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/7wgnecrs64j1n",
  "venue": "arXiv",
  "versions": []
}