Ground states for the NLS on non-compact graphs with an attractive potential

We consider the subcritical nonlinear Schr\"odinger equation on non-compact quantum graphs with an attractive potential supported in the compact core, and investigate the existence and the nonexistence of Ground States, defined as minimizers of the energy at fixed $L^2$-norm, or mass. We finally reach the following picture: for small and large mass there are Ground States. Moreover, according to the metric features of the compact core of the graph and to the strength of the potential, there may be an interval of intermediate masses for which there are no Ground States. The study was inspired by the research on quantum waveguides, in which the curvature of a thin tube induces an effective attractive potential.