The nonlinear Schrödinger equation on products of ${\mathbb{R}}^N$ and compact metric graphs

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 $2<p<2_{\ast }:=2+4\mathrm{/}(N+1)$. At the critical exponent $p=2_{\ast }$, 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 $2_{\ast }<p<2+4\mathrm{/}N$ 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 $\mathcal{G}$.