We consider the nonlinear Schrödinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the the graph is compact, for every value of the mass there is a constant solution. Our scope is to analyze (in dependence of the mass) the variational properties of this solution, as a critical point of the energy functional: local and global minimality, and (orbital) stability. We consider both the subcritical regime and the critical one, in which the features of the graph become relevant. We describe how the above properties change according to the topology and the metric properties of the graph.

中文翻译：

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紧缩度量图上NLS方程常数解的变分和稳定性

我们在一般紧缩度量图上考虑具有纯幂非线性的非线性Schrödinger方程，尤其是固定质量的平稳解。由于该图很紧凑，因此对于每个质量值都有一个恒定的解。我们的范围是分析（取决于质量）此解决方案的变化性质，作为能量函数的关键点：局部和全局最小值以及（轨道）稳定性。我们同时考虑了亚临界状态和临界状态，其中图的特征变得相关。我们将描述以上属性如何根据图的拓扑和度量属性进行更改。