Year: 2026
Type: preprint
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
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213499453