发表机构
Università degli Studi di Torino; Politecnico di Milano（都灵大学; 米兰理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究 $\mathbb{R}^N$ 与紧致度量图乘积上的质量约束聚焦非线性薛定谔方程，建立函数框架并证明基态存在性、临界阈值及维度交叉现象。
AI 中文摘要
我们在质量约束变分框架下，研究欧几里得空间 $\mathbb{R}^N$ 与紧致度量图 $\mathcal{G}$ 的乘积 $\mathbb{R}^N \times \mathcal{G}$ 上的稳态聚焦非线性薛定谔方程。该乘积是一种新型的混合结构：其所有面均为 $(N+1)$ 维，并沿余维为一的界面粘合，因此能量空间是真正的 Sobolev 空间，而图的度量与拓扑均进入变分问题。我们首先建立函数框架，给出 $H^1(\mathbb{R}^N \times \mathcal{G})$ 的两种等价描述，引入两个变量各自的偏重排及相应的 Pólya–Szegő 不等式，并证明局部形式的 Gagliardo–Nirenberg 不等式，且与 $\mathbb{R}^{N+1}$ 及半空间的最优常数进行比较。随后研究质量约束问题。当 $2<p<2_*:=2+4/(N+1)$ 时，对任意质量均存在基态。在临界指数 $p=2_*$ 时，基态存在于低于一个依赖于图的阈值之下，该阈值介于欧几里得临界质量的一半与完整欧几里得临界质量之间。当图存在环覆盖时，达到后者；而当存在末端边时，阈值恰好减半。两种情况下阈值均为尖锐的。对于 $2_*<p<2+4/N$，全局极小元不存在，但我们证明在进一步的质量阈值之下存在局部极小元，并给出其显式下界。最后，我们描述维度交叉：在临界质量之下，极小元不依赖于图变量，并刻画了半平凡解成为局部极小元的阈值，该阈值由 $\mathcal{G}$ 上 Kirchhoff 拉普拉斯算子的第一个非零特征值决定。
英文摘要
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 $\mathcal{G}$.
