The nonlinear Schrödinger equation on products of ℝ^N and compact metric graphs

@misc{soave2026nonlinear,
  title = {{The nonlinear Schrödinger equation on products of $\mathbbR^N$ and compact metric graphs}},
  author = {Nicola Soave and Gianmaria Verzini and Lorenzo Villata},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18698},
  eprint = {2609.18698},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18698},
}

TY  - UNPB
TI  - The nonlinear Schrödinger equation on products of ℝ^N and compact metric graphs
AU  - Nicola Soave
AU  - Gianmaria Verzini
AU  - Lorenzo Villata
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18698
UR  - https://arxiv.org/abs/2609.18698
AB  - We study the stationary focusing nonlinear Schrödinger equation on the product ℝ^N × 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(ℝ^N × 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 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 𝒢.
ER  -