{
  "abstract": "We investigate the existence and non-existence of anti-symmetric solutions to the fractional Lane--Emden system in the half-space $\\mathbb{R}^n_+$: \\begin{equation*}\\begin{cases} (-Δ)^s u = v^p & \\text{in } \\mathbb{R}^n_+, (-Δ)^s v = u^q & \\text{in } \\mathbb{R}^n_+, \\end{cases}\\end{equation*} where $0<s<\\frac{1}{2},p,q>0,2s+p=1$, with $u,v$ nonnegative in $\\mathbb{R}^n_+$ and anti-symmetric with respect to $\\partial\\mathbb{R}^n_+$. Li and Zhuo [2023, Sci. China Math.] previously proved non-existence in $L_{2s}$, as well as non-existence in the extended space $L_{2s+1}$ when $1-2s<p,q<\\frac{n+2s}{n-2s}$ and existence in $L_{2s+1}$ when $p+2s<1,q+2s<1$. In this paper, we fill the remaining borderline case $p=1-2s$ in $L_{2s+1}$ space. We establish a non-existence result for $1-2s=p\\leq q\\leq\\frac{n+2s}{n-2s}$ and an existence result for $0<q<p=1-2s$. For the non-existence result, we combine a maximum principle for anti-symmetric functions with the method of moving planes to derive a Liouville--type theorem when $1-2s<q\\leq\\frac{n+2s}{n-2s}$. We then iterate the lower-bound estimates to obtain a contradiction through blow-up. To prove the existence of nontrivial solutions when $0<q<1-2s$, we construct a nonhomogeneous solution by a monotone iteration scheme in $L_{2s+1}$ as well as analyze a homogeneous profile. Surprisingly, the homogeneous profile serves as a universal lower bound for all solutions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39165",
  "author_total": 2,
  "authors": [
    {
      "name": "Congming Li"
    },
    {
      "name": "Zixuan Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9sst5cyztrbmw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39165",
  "fulltext_url": null,
  "id": "9sst5cyztrbmw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39165",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39165",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9sst5cyztrbmw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39165",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39165"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Classification of Anti-symmetric Solutions to Fractional Lane-Emden System in Borderline Case",
  "title_text": "Classification of Anti-symmetric Solutions to Fractional Lane-Emden System in Borderline Case",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/9sst5cyztrbmw",
  "venue": "arXiv",
  "versions": []
}