{
  "abstract": "We study the symmetry, uniqueness, and nondegeneracy of positive solutions to the Choquard equation $-Δu+u=(I_α*u^p)u^{p-1}$ in $\\mathbb{R}^N$, which have long remained open problems. For $N\\geq3$, $α\\in(0,N)$, and $\\frac{N+α}{N}<p<\\frac{N+α}{N-2}$,we prove that every positive $H^1$-solution is radially symmetric and strictly decreasing in the radial variable. The symmetry proof relies on the moving-plane method, where the sublinear singularity is controlled by analyzing the far-field behavior of the nonlocal term. Uniqueness and nondegeneracy are established through a functional framework hinging on a nonnegative joint form and a Schur complement argument.Radial nondegeneracy and Morse index one are obtained by ruling out the spectral interval $(0,p-1]$ via a strict cumulative Riesz identity, which leads to uniqueness through a Nehari two-well argument. Nondegeneracy in the full space $H^1(\\mathbb{R}^N)$ follows from the spherical-harmonic decomposition of the linearized form. These results hold for the full range of $p$ when $α=2$, and for $p\\geq2$ when $α\\in(2,N)$. The cases $α\\in(0,2)$ and $α\\in(2,N)$ with $p<2$ remain open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05138",
  "author_total": 3,
  "authors": [
    {
      "name": "Jiankang Xia"
    },
    {
      "name": "Chengxiang Zhang"
    },
    {
      "name": "Xu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dkaj4fkrx9qne.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05138",
  "fulltext_url": null,
  "id": "dkaj4fkrx9qne",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05138",
  "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/2611.05138",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dkaj4fkrx9qne.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.05138",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05138"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Positive solutions to the nonlinear Choquard equation: symmetry, uniqueness, and nondegeneracy",
  "title_text": "Positive solutions to the nonlinear Choquard equation: symmetry, uniqueness, and nondegeneracy",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/dkaj4fkrx9qne",
  "venue": "arXiv",
  "versions": []
}