SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2874-2894, January 2020.
We consider the nonlinear Schrödinger equation with focusing power-type nonlinearity on compact graphs with at least one terminal edge, i.e., an edge ending with a vertex of degree 1. On the one hand, we introduce the associated action functional, and we provide a profile description of positive low action solutions at large frequencies, showing that they concentrate on one terminal edge, where they coincide with suitable rescaling of the unique solution to the corresponding problem on the half-line. On the other hand, a Lyapunov--Schmidt reduction procedure is performed to construct one-peaked and multipeaked positive solutions with sufficiently large frequency, exploiting the presence of one or more terminal edges.

中文翻译：

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带有终端边的图上NLS方程的峰值和低作用解

SIAM数学分析杂志，第52卷，第3期，第2874-2894页，2020年1月。
我们考虑具有幂次幂型非线性的非线性Schrödinger方程在具有至少一个终端边（即，一个以顶点结尾的边）的紧图上的度数。一方面，我们介绍了相关的作用函数，并提供了对大频率正向低作用解决方案的简要描述，表明它们集中于一个终端边缘，在此它们与独特解决方案的适当缩放相吻合在半线上对应的问题。另一方面，利用一个或多个末端边的存在，执行李雅普诺夫-施密特约简程序来构建频率足够大的单峰和多峰正解。