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

We establish general nonuniqueness results for normalized ground states of nonlinear Schrödinger equations with power nonlinearity on metric graphs. Basically, we show that, whenever in the 𝐿2-subcritical regime a graph hosts ground states at every mass, for nonlinearity powers close to the 𝐿2-critical exponent 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.