We consider the mass-critical non-linear Schrödinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in [R. Adami, E. Serra and P. Tilli. Negative energy ground states for the L2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387–406.] , where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved.

中文翻译：

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度量图上关键 NLS 能量在没有基态的情况下的局部最小化器

我们考虑非紧度量图上的质量临界非线性薛定谔方程。Adami、Serra 和 Tilli 在 [R. 阿达米、E. Serra 和 P. Tilli。L 的负能基态2- 度量图上的关键 NLSE。通讯。数学。物理。352 (2017), 387–406.] ，其中证明基态的存在和性质在关键的方式上取决于质量的值和基础图的拓扑性质。在本文中，我们解决了不存在基态的情况，并表明，在适当的假设下，能量的受限局部最小化器确实存在。在未达到基态能级的情况下，该结果为时间相关方程中存在稳定解铺平了道路。