{
  "abstract": "In this work, we establish a sharp logarithmic Choquard inequality by combining Beckner's entropy inequality with the sharp Pitt-type inequality. Motivated by this estimate, we classify the positive classical solutions, in a suitable integrability class, of the critical logarithmic Choquard problem \\[ \\mathcal{L}_Δu = σu + \\frac{1}{\\|u\\|_2^2} \\left(G_{\\ln}(u)-\\frac{4}{N}\\int_{\\mathbb{R}^N}u^2\\ln u\\,dx\\right)u \\qquad\\text{in }\\mathbb{R}^N, \\] where $σ\\in\\mathbb{R}$, $N\\geq 1$, $\\mathcal{L}_Δ$ is the logarithmic Laplacian, and $G_{\\ln}(u)=\\ln(1/|x|^4)*u^2$. We construct explicit solutions as limits of critical fractional Choquard bubbles and prove that every positive classical solution in the prescribed class has the form \\[ u_{σ,t}(x) = e^{\\frac{N}{4}(σ-B_{N,\\mathcal{L}})} B_{N,0}\\left(\\frac{t}{t^2+|x-x_0|^2}\\right)^{N/2}, \\qquad t>0,\\quad x_0\\in\\mathbb{R}^N, \\] where \\[ B_{N,0} =\\left(\\frac{Γ(N)}{π^{N/2}Γ(N/2)}\\right)^{1/2}, \\qquad B_{N,\\mathcal{L}} =2\\ln 2+4ψ(N/2)-2ψ(N)+\\frac{4}{N}\\ln B_{N,0}. \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35669",
  "author_total": 2,
  "authors": [
    {
      "name": "Rakesh Arora"
    },
    {
      "name": "Jacques Giacomoni"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2005jycnhzf46.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35669",
  "fulltext_url": null,
  "id": "2005jycnhzf46",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35669",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214789023",
  "pdf_url": "https://arxiv.org/pdf/2609.35669",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2005jycnhzf46.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35669",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35669"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the classification of solutions to the Logarithmic Laplacian critical Choquard equation",
  "title_text": "On the classification of solutions to the Logarithmic Laplacian critical Choquard equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/2005jycnhzf46",
  "venue": "arXiv",
  "versions": []
}