{
  "abstract": "We establish sharp finite-ball shooting classifications for radial nodal Dirichlet solutions of the logarithmic equation and, within the maximal simple-zero shooting class, of the sublinear scalar-field equation. Recent whole-space uniqueness and phase-transition theorems are used as external inputs, while the finite-ball zero-curve ranges, endpoint asymptotics, and spectral consequences are proved here. The two models require different singular analyses: in the logarithmic problem the nonlinearity is not locally Lipschitz at a nodal zero and the linearized potential diverges there, whereas in the sublinear problem the limiting whole-space profile reaches a double zero at a finite support radius, beyond which continuation is nonunique. For every $R>0$ and $k\\ge0$, the logarithmic problem has, up to sign, a unique radial Dirichlet solution with exactly $k$ interior zeros. Its shooting height $β_k^{\\log}(R)$ is a strictly decreasing $C^1$ bijection from $(0,\\infty)$ onto $(α_k^{\\log},\\infty)$ and satisfies \\[ β_k^{\\log}(R)\\longrightarrowα_k^{\\log} \\quad(R\\to\\infty),\\qquad \\log\\bigl(β_k^{\\log}(R)^2\\bigr) =\\frac{ρ_{k+1}^2}{R^2}+κ_{k+1,n}+o(1) \\quad(R\\downarrow0), \\] where $κ_{k+1,n}>0$ is given by an explicit Bessel integral. For the sublinear problem, the maximal simple-zero branch exists precisely for $R\\in(ρ_{k+1},S_k)$, where $ρ_{k+1}$ is the $(k+1)$-st positive zero of the regular Bessel profile $Φ_n$, and $S_k$ is the support radius of the unique compactly supported $k$-node whole-space bound state.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15283",
  "author_total": 2,
  "authors": [
    {
      "name": "Wenjing Chen"
    },
    {
      "name": "Zexi Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/82zw7hmwy195p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15283",
  "fulltext_url": null,
  "id": "82zw7hmwy195p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15283",
  "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": "W7213259811",
  "pdf_url": "https://arxiv.org/pdf/2609.15283",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/82zw7hmwy195p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15283",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15283"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Radial Nodal Dirichlet Solutions of Singular Elliptic Equations: Global Branches, Endpoint Asymptotics, and Morse Indices",
  "title_text": "Radial Nodal Dirichlet Solutions of Singular Elliptic Equations: Global Branches, Endpoint Asymptotics, and Morse Indices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/82zw7hmwy195p",
  "venue": "arXiv",
  "versions": []
}