{
  "abstract": "Let $N>p>1$, $s>-p$, $0<μ<\\barμ:=\\big(\\frac{N-p}{p}\\big)^{p}$ and $q>p-1$. We determine the behaviour at infinity of \\emph{every} positive solution of the quasilinear Hardy--Emden--Fowler equation \\[ -\\big(r^{N-1}|v'|^{p-2}v'\\big)'=μ\\,r^{N-1-p}v^{p-1}+r^{N-1+s}v^{q}, \\qquad r>R_{0}. \\] Three rates compete: the self-similar rate $r^{-γ_{*}}$ with $γ_{*}=\\frac{p+s}{q-p+1}$, and the two Hardy rates $r^{-ν_{\\pm}}$, where $ν_{\\pm}$ are the roots of $h(ν):=ν^{p-1}(N-p-(p-1)ν)=μ$. Writing $p^{*}_{s}=\\frac{p(N+s)}{N-p}$ and $q_{S}=p-1+\\frac{p+s}{ν_{+}}$ for the Hardy--Sobolev and Serrin exponents, we obtain a complete picture in four regimes. If $q>p^{*}_{s}-1$, every solution realises exactly one of the three rates, with a logarithmic correction in the borderline case $h(γ_{*})=μ$, and we compute the leading constants. If $q=p^{*}_{s}-1$ the system becomes integrable: besides the singular self-similar solution and a heteroclinic ``bubble'' there is a one-parameter family of Fowler--Delaunay solutions, for which $r^{γ_{*}}v(r)$ is a non-constant periodic function of $\\ln r$. If $q_{S}<q<p^{*}_{s}-1$ only two orbits survive, and if $q\\le q_{S}$ there is no positive solution at all in an exterior domain. This is the quasilinear, weighted counterpart of the picture obtained by Franca and Sfecci for $p=2$. We work with the slope variables $x=-rv'/v$, $Y=r^{p+s}v^{q-p+1}$, in which the equation becomes a planar system that is real-analytic on the half-plane $\\{x>0\\}$ for every $1<p<N$, so that no restriction of the type $p\\le2$ is needed; the analysis then rests on a universal barrier $x<ν_{+}$, on an explicit Dulac function which degenerates into a first integral exactly at $q=p^{*}_{s}-1$, and on an explicit Lyapunov function.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09724",
  "author_total": 1,
  "authors": [
    {
      "name": "Phuong Le"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0p176f97y64f9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09724",
  "fulltext_url": null,
  "id": "0p176f97y64f9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09724",
  "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": "W7212178037",
  "pdf_url": "https://arxiv.org/pdf/2609.09724",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0p176f97y64f9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09724",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09724"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Radial solutions of quasilinear Hardy--Emden--Fowler equations: exact decay rates, Delaunay solutions, and nonexistence",
  "title_text": "Radial solutions of quasilinear Hardy--Emden--Fowler equations: exact decay rates, Delaunay solutions, and nonexistence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/0p176f97y64f9",
  "venue": "arXiv",
  "versions": []
}