{
  "abstract": "We investigate radial bound states with prescribed nodes for the odd cubic equation \\[ Δu+u(\\abs{u}-b)(c-\\abs{u})=0 \\quad\\text{in }\\R^n,\\qquad n\\ge2,\\quad 0<b<c. \\] Writing $a=b/c$, we prove that a radial bound state with exactly $k\\ge1$ simple zeros and positive central value exists if and only if $0<a<1/2$. For each $k$, this solution is unique for all $0<a<1/2$ in dimension two, for $a\\ge(7-3\\sqrt5)/2$ in dimension three, and for $a\\ge(43-5\\sqrt{33})/32$ in dimension four. For one node, we identify exact ranges obtained from a tangent comparison and a quotient inequality in every dimension $n\\ge3$; the threshold is dimension-independent for $3\\le n\\le23$ and is characterized implicitly for $n\\ge24$. These results address the natural odd counterpart of the cubic nonlinearity highlighted in Tang's prescribed-node uniqueness conjecture [M. Tang, \\emph{Invent. Math.}, 2026]. The main challenges are the second positive zero of the cubic source, the lack of a global comparison inequality, and supercritical cubic growth in higher dimensions. We overcome them by an amplitude confinement argument, controlled extensions preserving sharp comparison estimates, and a subcritical truncation for existence. A complementary small-$a$ uniqueness range in dimensions three to five follows from a recent massive double-power result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34025",
  "author_total": 2,
  "authors": [
    {
      "name": "Wenjing Chen"
    },
    {
      "name": "Zexi Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fzahzrf8s9b5d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34025",
  "fulltext_url": null,
  "id": "fzahzrf8s9b5d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34025",
  "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": "W7223799653",
  "pdf_url": "https://arxiv.org/pdf/2610.34025",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fzahzrf8s9b5d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34025",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34025"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Uniqueness of radial bound states for an equation with an odd cubic-like nonlinearity",
  "title_text": "Uniqueness of radial bound states for an equation with an odd cubic-like nonlinearity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/fzahzrf8s9b5d",
  "venue": "arXiv",
  "versions": []
}