{
  "abstract": "We study the critical fractional $p$-Hardy-Sobolev equation \\begin{equation}\\tag{$\\mathcal{P}$}\\label{a-main} (-Δ_p)^s u -μ\\dfrac{|u|^{p-2}u}{|x|^{sp}}=\\dfrac{|u|^{p^*_s(α)-2}u}{|x|^α}+f \\;\\mbox{ in }\\,\\mathbb{R}^d, \\quad u\\in \\mathcal{D}^{s,p}(\\mathbb{R}^d), \\end{equation} where $1<p<\\infty$, $0<s<1$, $0\\leqα<sp<d$, $μ>0$, $p^*_s(α):= p(d-α)/(d-sp)$ is the critical Hardy-Sobolev exponent, and $f$ is a nontrivial nonnegative functional in $(\\mathcal{D}^{s,p}(\\mathbb{R}^d))^*$. We first establish global compactness results for Palais-Smale sequences associated with the corresponding energy functional. When $α>0$, the loss of compactness is described by dilations of solutions of the Hardy-Sobolev limit problem. The case $α=0$ has a different structure: in addition to Hardy profiles, pure Sobolev profiles may occur when the centre of concentration escapes from the Hardy singularity relative to its scale. We give a direct centre-scale analysis of these two concentration regimes and obtain the corresponding energy decomposition and profile separation. As an application, under an explicit smallness assumption on $f$, we first obtain a positive solution for \\eqref{a-main} with negative energy. We then construct a nonlinear path based on hidden convexity whose energy remains strictly below the first bubbling threshold. A minimax argument, combined with the global compactness theorem, then yields a second distinct positive solution for \\eqref{a-main}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08510",
  "author_total": 3,
  "authors": [
    {
      "name": "Nirjan Biswas"
    },
    {
      "name": "Souptik Chakraborty"
    },
    {
      "name": "Debangana Mukherjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1gchza5g9hf31.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08510",
  "fulltext_url": null,
  "id": "1gchza5g9hf31",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08510",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7212013459",
  "pdf_url": "https://arxiv.org/pdf/2609.08510",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1gchza5g9hf31.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08510",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08510"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Critical fractional $p$-Hardy Sobolev equations: Global compactness and multiplicity of positive solutions",
  "title_text": "Critical fractional p-Hardy Sobolev equations: Global compactness and multiplicity of positive solutions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/1gchza5g9hf31",
  "venue": "arXiv",
  "versions": []
}