Existence of infinite stationary solutions of the $L^2$-subcritical and critical NLSE on compact metric graphs

TL;DR

This paper proves the existence of infinitely many stationary solutions for the nonlinear Schrödinger equation on compact metric graphs in the subcritical case, and characterizes the solution set in the critical case based on mass and graph topology.

Contribution

It establishes the existence of infinite solutions in the subcritical regime and characterizes the critical regime solutions relative to mass and topology, using variational methods.

Findings

Infinite stationary solutions in the subcritical case.

Solution existence threshold related to graph topology.

Characterization of solution set in the critical case.

Abstract

We investigate the existence of stationary solutions for the Nonlinear Schr\"odinger equation on compact metric graphs. In the L2-subcritical setting, we prove the existence of an infinite number of such solutions, for every value of the mass. In the critical regime, this infinity of solutions is established to exists if and only if the mass is lower or equal to a threshold value. Moreover, the relation between this threshold and the topology of the graph is characterized. The investigation is based on variational techniques and some new versions of Gagliardo-Nirenberg inequalities.