{
  "abstract": "We study Hamiltonian elliptic systems with prescribed bilinear mass $\\int_{\\mathbb{R}^N}uv=a>0$. We establish a Gagliardo--Nirenberg inequality in the crossed gradient pairing and bilinear mass, on scaling-invariant component-bounded classes. It identifies the critical curve $1/p+1/q=N/(N+2)$ and the restricted energy trichotomy. For power nonlinearities, we establish uniqueness up to common translations and nondegeneracy of positive profiles throughout the Sobolev-subcritical hyperbola. Exact scaling then classifies positive normalised solutions and identifies the unique critical mass. In dimension two, we develop a bilinear exponential Gagliardo--Nirenberg estimate with a sharp gradient threshold and a truncated scalar refinement with the optimal quartic leading coefficient. Adapting variational ideas from [Cassani-Tarsi, Calc.Var.PDE (2015)] we combine a mass-normalising quotient, reduction over the full negative fibres and compactness below the concentration threshold to obtain least-energy positive radial solutions for pure exponential nonlinearities at every prescribed mass below the cubic limiting mass. Local perturbative branches and a small-frequency exponential branch yield explicit mass-response formulas.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08774",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniele Cassani"
    },
    {
      "name": "Giulio Romani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5fpsvgypeejj9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08774",
  "fulltext_url": null,
  "id": "5fpsvgypeejj9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08774",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7220911189",
  "pdf_url": "https://arxiv.org/pdf/2610.08774",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5fpsvgypeejj9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.08774",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08774"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Normalised Hamiltonian Elliptic Systems: a Gagliardo-Nirenberg inequality for bilinear mass",
  "title_text": "Normalised Hamiltonian Elliptic Systems: a Gagliardo-Nirenberg inequality for bilinear mass",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/5fpsvgypeejj9",
  "venue": "arXiv",
  "versions": []
}