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

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.