{
  "abstract": "We study the Brezis--Nirenberg problem for the fractional Laplacian with an upper critical Hartree nonlinearity on a bounded domain $Ω\\subset\\mathbb R^N$, in the low-dimensional range $2s<N<4s$. For a continuous potential $a$, we prove that the following properties are equivalent: the Robin function of $(-Δ)^s+a$ is negative somewhere in $Ω$; the associated Hartree--Sobolev quotient ${\\mathcal S_{H,s}}(a)$ lies strictly below the best constant ${\\mathcal S_{H,s}}$; and ${\\mathcal S_{H,s}}(a)$ is attained. This leads to the notion of a critical potential, for which ${\\mathcal S_{H,s}}(a)={\\mathcal S_{H,s}}$ and every nontrivial decrease of $a$ lowers the infimum. For a critical potential $a$ and $V\\in L^\\infty(Ω)$, we determine the leading term in the asymptotics of ${\\mathcal S_{H,s}}(a+\\varepsilon V)$ as $\\varepsilon\\to0^+$ and describe the blow-up of almost-minimizers, including the concentration speed and the location of the concentration point. Compared with the Sobolev case, the nonlocal nonlinearity produces several new interaction terms; a key ingredient is a proof of non-attainment at the threshold based on the sharp Hardy--Littlewood--Sobolev inequality on the sphere.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26161",
  "author_total": 3,
  "authors": [
    {
      "name": "Giusi Vaira"
    },
    {
      "name": "Yue Wu"
    },
    {
      "name": "Minbo Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7sdng7sk4r7jx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26161",
  "fulltext_url": null,
  "id": "7sdng7sk4r7jx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26161",
  "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": "W7223050079",
  "pdf_url": "https://arxiv.org/pdf/2610.26161",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7sdng7sk4r7jx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26161",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26161"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Critical functions and asymptotics for fractional Hartree Brezis--Nirenberg problems in low dimensions",
  "title_text": "Critical functions and asymptotics for fractional Hartree Brezis--Nirenberg problems in low dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/7sdng7sk4r7jx",
  "venue": "arXiv",
  "versions": []
}