{
  "abstract": "Let $n\\ge2$, $0<s<1$, and $p,q>0$. We prove a Liouville theorem for the fractional Lane--Emden system \\[ (-Δ)^s u=v^p,\\qquad (-Δ)^s v=u^q, \\qquad u, v>0 \\qquad\\text{in }\\R^n. \\] More precisely, we show that the system does not have a strictly positive entire solution whenever \\[ \\frac1{p+1}+\\frac1{q+1}>\\frac{n-2s}{n}. \\] No radial symmetry, global boundedness, finite-energy assumption, or prescribed decay at infinity is required. Our proof adapts the localized potential-kernel virial and source-layer interval-pressure method recently introduced for the classical Hénon--Lane--Emden system. The basic reason the method also works in the fractional setting is simple: the Riesz kernel associated with $(-Δ)^s$ has the same logarithmic derivative structure as the Newton kernel, with $n-2$ replaced by $n-2s$. However, two genuinely nonlocal points require special treatments. When $2s\\le1$, the individual kernel-gradient terms are not locally absolutely integrable, so the two equations must be combined before the diagonal limit is taken. When $2s>n-1$, the kernel restricted to a line is not integrable at infinity, and the interval comparison must therefore be formulated using nonnegative extended integrals. After these issues are resolved, a localized virial identity, a one-dimensional interval-pair inequality, and an averaged source-layer estimate yield a universal local energy bound. The subcritical scaling then forces the same energy to diverge, giving the desired contradiction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10188",
  "author_total": 2,
  "authors": [
    {
      "name": "Linfeng Mei"
    },
    {
      "name": "Juncheng Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/02nrkn35rk80q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10188",
  "fulltext_url": null,
  "id": "02nrkn35rk80q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10188",
  "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": "W7220939748",
  "pdf_url": "https://arxiv.org/pdf/2610.10188",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/02nrkn35rk80q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10188",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10188"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A proof of the fractional Lane--Emden conjecture",
  "title_text": "A proof of the fractional Lane--Emden conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/02nrkn35rk80q",
  "venue": "arXiv",
  "versions": []
}