{
  "abstract": "We show the existence of a ``nonradial\" smooth entire solution of the Ginzburg--Landau equation with degree $q\\in\\{8,9\\}$ and modulus tending to one at infinity. This answers Brezis' Open Problem 2.4 negatively. To construct the counterexample, we vary a cone parameter $q \\in [8, 9]$ and keeping the degree on the cone equal to one. Several key ingredients are needed. First, we need to compute the Morse index of radial vortex solutions at $q\\in \\{8, 9\\}$. Second, we need to show compactness. Finally, Leray-Schauder degree theory and a homotopy argument yield the desired existence result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34979",
  "author_total": 4,
  "authors": [
    {
      "name": "Hong-Ge Chen"
    },
    {
      "name": "Yong Liu"
    },
    {
      "name": "Juncheng Wei"
    },
    {
      "name": "Wen Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8jf22ddyf2a40.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34979",
  "fulltext_url": null,
  "id": "8jf22ddyf2a40",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34979",
  "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": "W7223856512",
  "pdf_url": "https://arxiv.org/pdf/2610.34979",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8jf22ddyf2a40.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34979",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34979"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On Brezis' open problem 2.4",
  "title_text": "On Brezis' open problem 2.4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/8jf22ddyf2a40",
  "venue": "arXiv",
  "versions": []
}