Doubly nonlinear Schr\"odinger ground states on metric graphs

TL;DR

This paper studies the existence of ground states for doubly nonlinear Schrödinger equations on metric graphs, revealing complex dependencies on both the graph's topology and metric, and extending previous analyses of standard nonlinearities.

Contribution

It provides a comprehensive topological and metric characterization of ground state existence for doubly nonlinear Schrödinger equations on metric graphs, highlighting richer phenomena than in previous single-nonlinearity cases.

Findings

Topological features determine ground state existence in specific mass regimes.

The metric influences the interplay between topology and ground state existence.

The phenomenology of doubly nonlinear cases is richer than in standard nonlinear cases.

Abstract

We investigate the existence of ground states at prescribed mass on general metric graphs with half-lines for focusing doubly nonlinear Schr\"odinger equations involving both a standard power nonlinearity and delta nonlinearities located at the vertices. The problem is proved to be sensitive both to the topology and to the metric of the graph and to exhibit a phenomenology richer than in the case of the sole standard nonlinearity considered in [Adami et al. '15, Adami et al. '16]. On the one hand, we provide a complete topological characterization of the problem, identifying various topological features responsible for existence/non-existence of doubly nonlinear ground states in specific mass regimes. On the other hand, we describe the role of the metric in determining the exact interplay between these different topological properties.