{
  "abstract": "We study positive classical solutions of the polyharmonic Hénon--Lane--Emden system $(-Δ)^m u=|x|^αv^p$, $(-Δ)^m v=|x|^βu^q$ in $\\mathbb{R}^n$, where $m\\ge2$, $p,q\\ge1$, $pq>1$, and $α,β\\ge0$. We prove that there is no such solution whenever $\\frac{n+α}{p+1}+\\frac{n+β}{q+1}>n-2m$, in every dimension and without any assumption on the behavior of the solutions at infinity. Combined with an existence result of Villavert, this proves the polyharmonic Hénon--Lane--Emden conjecture of Phan for $m\\ge2$: for these exponents and weights, a positive classical solution exists if and only if $\\frac{n+α}{p+1}+\\frac{n+β}{q+1}\\le n-2m$. The proof has two parts. We first show that positivity forces $(-Δ)^i u>0$ and $(-Δ)^i v>0$ for $1\\le i\\le m-1$. For $m$ even this needs $pq>1$, and explicit radial examples show that it fails when $pq<1$ and when $p=q=1$. The system is then equivalent to a weighted integral system with a Riesz kernel of order $2m$. For such integral systems we prove the corresponding Liouville theorem for every Riesz order $μ\\in(1,n)$, adapting a recent argument for the second-order case that combines a localized virial identity with an interval comparison along source layers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02448",
  "author_total": 1,
  "authors": [
    {
      "name": "Phuong Le"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a7mjbrpv0n9th.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02448",
  "fulltext_url": null,
  "id": "a7mjbrpv0n9th",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02448",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02448",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a7mjbrpv0n9th.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02448",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02448"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The polyharmonic Hénon--Lane--Emden conjecture",
  "title_text": "The polyharmonic Hénon--Lane--Emden conjecture",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/a7mjbrpv0n9th",
  "venue": "arXiv",
  "versions": []
}