Nonuniqueness of normalized ground states for nonlinear Schrödinger equations on metric graphs

We establish general nonuniqueness results for normalized ground states of nonlinear Schr & ouml;dinger equations with power nonlinearity on metric graphs. Basically, we show that, whenever in the L2$L<^>2$-subcritical regime a graph hosts ground states at every mass, for nonlinearity powers close to the L2$L<^>2$-critical exponent p=6$p=6$, there is at least one value of the mass for which ground states are nonunique. As a consequence, we also show that, for all such graphs and nonlinearities, there exist action ground states that are not normalized ground states.