{
  "abstract": "We consider the Brezis--Nirenberg problem $-Δu=|u|^{2^*-2}u+\\varepsilon u$ in the unit ball of $\\mathbb R^N$, $N\\ge5$, with Dirichlet boundary conditions, and we look for sign-changing solutions which, as $\\varepsilon\\to0^+$, blow up positively at the centre and negatively at the $k$ vertices of a regular polygon. The key observation is that, for this configuration, the gradient of the reduced energy factorises, so that the whole finite-dimensional problem is governed by the zeros of a single explicit function of the radius of the polygon. Its behaviour depends on the dimension: when $N=5$ such solutions exist for every $k\\ge2$, and the ring of negative peaks approaches the boundary at distance $O(1/k)$; when $N\\ge6$ there are two solutions for $2\\le k\\le6$, while no solution of this type exists for $k$ large. For $2\\le k\\le6$ the nodal set of every such solution reaches the boundary of the ball, in contrast with what happens for the almost critical problem. Nevertheless, the two solutions found for $N\\ge6$ have very different nodal structures: numerically, for $3\\le k\\le6$ one of them has only two nodal domains, and its nodal set meets the plane of the peaks along a closed curve around the centre, while the other one can have as many as $k+1$ nodal domains. These results extend to higher dimensions a configuration previously found in dimension four.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26107",
  "author_total": 3,
  "authors": [
    {
      "name": "Angela Pistoia"
    },
    {
      "name": "Giuseppe Mario Rago"
    },
    {
      "name": "Giusi Vaira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0j2azrm6rjrwb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26107",
  "fulltext_url": null,
  "id": "0j2azrm6rjrwb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26107",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7223045863",
  "pdf_url": "https://arxiv.org/pdf/2610.26107",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0j2azrm6rjrwb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26107",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26107"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "One bubble against a ring: new sign-changing solutions of the Brezis-Nirenberg problem",
  "title_text": "One bubble against a ring: new sign-changing solutions of the Brezis-Nirenberg problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/0j2azrm6rjrwb",
  "venue": "arXiv",
  "versions": []
}