{
  "abstract": "We study ground states, criticality, and variational compactness for genuinely multipolar $L^p$-Hardy weights, $1<p<n$. A ground-state representation across the poles gives a complete criticality classification of the full potential and characterizes finite-energy attainment. Removing a geometric interaction term produces a variational problem governed by concentration at the poles and infinity. A strict concentration gap yields compactness, positive extremizers, and criticality of the optimal operator; for $p\\ge2$, it is also necessary for attainment. In the subquadratic range, the optimal truncated coefficient is below one and attained throughout the full-potential critical set. For admissible parameters with $1<p<2$ and $(n-p)/p<s\\le m(n-p)/p$, arbitrary fixed pole configurations exhibit negative directions, attained critical states, and coercive subcritical regimes outside this set along positive-weight paths towards the simplex vertices. A single-pole degeneration formula and continuity in the weights establish this coexistence. In the superquadratic range, an interaction threshold determines a non-attained infinity-concentration interval. Under strict separation from pole concentration, its endpoint is critical with infinite energy and a compact phase follows. We prove separation for every bipolar configuration and in several multipolar regimes, including a vertex neighborhood uniform in the exponent. Near parameters strictly inside the quadratic attained region, the optimal coefficient crosses one continuously through $p=2$. The proofs combine ground-state remainders, capacity, weighted compactness, and critical Hardy barriers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00652",
  "author_total": 3,
  "authors": [
    {
      "name": "Yongyang Jin"
    },
    {
      "name": "Shoufeng Shen"
    },
    {
      "name": "Li Tang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4w1zm9cj86j42.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00652",
  "fulltext_url": null,
  "id": "4w1zm9cj86j42",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00652",
  "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",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7216195845",
  "pdf_url": "https://arxiv.org/pdf/2610.00652",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4w1zm9cj86j42.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.00652",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00652"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Optimal multipolar Lp-Hardy inequalities: ground states, criticality and variational thresholds",
  "title_text": "Optimal multipolar Lp-Hardy inequalities: ground states, criticality and variational thresholds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/4w1zm9cj86j42",
  "venue": "arXiv",
  "versions": []
}