{
  "abstract": "We obtain new results on Lane-Emden conjecture to prove a Liouville theorem for the Lane-Emden system $$\\begin{cases} -Δu=v^p,& x\\in \\mathbb{R}^N,\\\\ -Δv=u^q,& x\\in \\mathbb{R}^N,\\end{cases}$$ with $N\\ge3$, $p,q>0$, $pq>1$, and \\[ \\left(p-\\frac2{N-2}\\right)^2+ \\left(q-\\frac2{N-2}\\right)^2\\le\\frac{2N^2}{(N-2)^2}, \\] the system has no positive classical solution as $(p,q)$ in the disk unless $p=q=(N+2)/(N-2)$, no growth or decay assumptions are imposed, and the disk is tangent to the critical hyperbola. Especially, for $N\\ge5$, the disk contains a nonempty open set not covered by the criteria of Busca-Manásevich~\\cite{BM} and Souplet~\\cite{Souplet}, or by the explicit conditions of Li-Li-Wei~\\cite{LLW}. After reducing the exponent range by known Liouville criteria, we first construct vector fields to derive a divergence inequality, then apply Sobolev embedding theorem, Gagliardo-Nirenberg interpolation inequality, and Young's inequality to give a uniform local integral bound $\\int_{B_1}(u^rv^{2p}+u^{2q}v^s)dx\\le C(N,p,q)$ for suitable weights for any positive solution $(u,v)$, finally the nonexistence of positive classical solutions follows by scaling.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09526",
  "author_total": 2,
  "authors": [
    {
      "name": "Haoyang Lu"
    },
    {
      "name": "Zhitao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45sv0dyq9n80j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09526",
  "fulltext_url": null,
  "id": "45sv0dyq9n80j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09526",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7220983772",
  "pdf_url": "https://arxiv.org/pdf/2610.09526",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45sv0dyq9n80j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09526",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09526"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "New result on Lane-Emden conjecture with exponents in a disk",
  "title_text": "New result on Lane-Emden conjecture with exponents in a disk",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/45sv0dyq9n80j",
  "venue": "arXiv",
  "versions": []
}