{
  "abstract": "We investigate positive radial solutions with an isolated nonremovable singularity for the semilinear elliptic equation \\begin{align*} Δ_{\\mathbb{H}^N} u+λu+u^p=0 \\qquad\\text{in }\\mathbb{H}^N\\setminus\\{Q\\}, \\end{align*} where $N\\geq 3$, $p>1$, $λ\\le \\frac{(N-1)^2}{4}$, and $Q\\in\\mathbb{H}^N$ is a prescribed pole. Our purpose is to describe how the locally Euclidean singular behavior near ${\\{Q}\\}$ interacts with the genuinely hyperbolic dynamics at infinity, and how this interaction changes across the Serrin and Sobolev critical exponents. For $\\frac{N}{N-2}\\le p<\\frac{N+2}{N-2}$, we construct a family of positive radial singular solutions selecting the fast exponential mode at infinity. At the pole, these solutions exhibit the logarithmically corrected fundamental-solution profile when $p=\\frac{N}{N-2}$, and the standard power-law profile when $\\frac{N}{N-2}<p<\\frac{N+2}{N-2}.$ At the conformally invariant pair $(p,λ)=(\\frac{N+2}{N-2},\\frac{N(N-2)}{4})$, we obtain an explicit singular solution and nonconstant Fowler-type solutions, and we classify all positive radial solutions with a nonremovable singularity via the positive periodic orbits of an associated autonomous equation. In the supercritical regime $p>\\frac{N+2}{N-2}$ with $λ\\le\\frac{ N(N-2)}{4}$, we prove existence and uniqueness of the global positive radial singular solution, derive its two-term local asymptotic expansion near the pole, and establish a sharp trichotomy for its behavior at infinity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06113",
  "author_total": 3,
  "authors": [
    {
      "name": "Xia Huang"
    },
    {
      "name": "Yahui Jiang"
    },
    {
      "name": "Chunyi Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/39c20kd9j2z3p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06113",
  "fulltext_url": null,
  "id": "39c20kd9j2z3p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06113",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.06113",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/39c20kd9j2z3p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06113",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06113"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Qualitative analysis of positive radial singular solutions on hyperbolic space",
  "title_text": "Qualitative analysis of positive radial singular solutions on hyperbolic space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/39c20kd9j2z3p",
  "venue": "arXiv",
  "versions": []
}