{
  "abstract": "We study the stationary focusing nonlinear Schrödinger equation on the product $\\mathbb{R}^N \\times \\mathcal{G}$ of the Euclidean space with a compact metric graph, in the mass-constrained variational setting. Such a product is a hybrid structure of a new type: all its faces are $(N+1)$-dimensional and are glued along interfaces of codimension one, so that the energy space is a genuine Sobolev space, while both the metric and the topology of the graph enter the variational problem. We first develop the functional framework, giving two equivalent descriptions of $H^1(\\mathbb{R}^N \\times \\mathcal{G})$, introducing partial rearrangements in each of the two variables together with the corresponding Pólya--Szegő inequalities, and proving Gagliardo--Nirenberg inequalities in a localized form, with a comparison of the optimal constants with those of $\\mathbb{R}^{N+1}$ and of the half-space. We then study the mass-constrained problem. Ground states exist for every mass when $2<p<2_*:=2+4/(N+1)$. At the critical exponent $p=2_*$, they exist below a graph-dependent threshold lying between one half of the Euclidean critical mass and the full Euclidean critical mass. The latter value is attained when the graph admits a cycle covering, whereas the threshold is exactly halved in the presence of a terminal edge. In both cases, the threshold is sharp. For $2_*<p<2+4/N$ global minimizers do not exist, but we prove the existence of local minimizers below a further mass threshold, for which we give an explicit lower bound. Finally, we describe the dimensional crossover: below a critical mass the minimizers do not depend on the graph variable, and we characterize the threshold below which the semi-trivial solution is a local minimizer in terms of the first nonzero eigenvalue of the Kirchhoff Laplacian on $\\mathcal{G}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18698",
  "author_total": 3,
  "authors": [
    {
      "name": "Nicola Soave"
    },
    {
      "name": "Gianmaria Verzini"
    },
    {
      "name": "Lorenzo Villata"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ej3hsv0qnqp8d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18698",
  "fulltext_url": null,
  "id": "ej3hsv0qnqp8d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18698",
  "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-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213499453",
  "pdf_url": "https://arxiv.org/pdf/2609.18698",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ej3hsv0qnqp8d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:03Z",
      "native_id": "2609.18698",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18698"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "The nonlinear Schrödinger equation on products of $\\mathbb{R}^N$ and compact metric graphs",
  "title_text": "The nonlinear Schrödinger equation on products of ℝ^N and compact metric graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/ej3hsv0qnqp8d",
  "venue": "arXiv",
  "versions": []
}