# Peaked and low action solutions of NLS equations on graphs with terminal edges

## TL;DR

This paper studies the nonlinear Schrödinger equation on graphs with terminal edges, describing low action solutions concentrating on edges and constructing multi-peaked solutions at high frequencies.

## Contribution

It introduces a profile description for low action solutions and develops a Ljapunov-Schmidt reduction to construct multi-peaked solutions on graphs with terminal edges.

## Key findings

- Low action solutions concentrate on terminal edges.
- Existence of multi-peaked solutions at large frequencies.
- Solutions resemble rescaled real-line solutions.

## Abstract

We consider the nonlinear Schr\"odinger 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 real line. On the other hand, a Ljapunov-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.