Existence of infinitely many stationary solutions of the L2-subcritical and critical NLSE on compact metric graphs

Abstract

We investigate the existence of stationary solutions for the nonlinear Schrödinger 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, the existence of infinitely many solutions is established if the mass is lower than a threshold value, while global minimizers of the NLS energy exist if and only if the mass is lower or equal to the threshold. 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.