Ground states for the NLS equation with combined nonlinearities on non-compact metric graphs

TL;DR

This paper studies the existence of ground states for the nonlinear Schrödinger equation with combined nonlinearities on non-compact metric graphs, revealing how graph topology and metric properties influence solutions.

Contribution

It introduces analysis of combined critical and subcritical nonlinearities on graphs, highlighting new phenomena and the impact of graph structure on ground state existence.

Findings

Graph topology significantly affects ground state existence.

Combined nonlinearities lead to new phenomena not seen in pure cases.

Topological and metric properties influence solution existence and non-existence.

Abstract

We investigate the existence of ground states with prescribed mass for the NLS energy with combined $L^2$-critical and subcritical nonlinearities, on a general non-compact metric graph $\mathcal{G}$. The interplay between the different nonlinearities creates new phenomena with respect to purely critical or subcritical problems on graphs; from a different perspective, topological and metric properties of the underlying graph drastically influence existence and non-existence of ground states with respect to the analogue problem on the real line.