Nonlinear Schrödinger equations on branched structures: the role of topology in the existence of ground states

We describe some results on the existence or nonexistence of ground states of prescribed mass for the nonlinear Schrödinger equation on noncompact metric graphs. We will highlight the role played by the topology of the graph in all the existence results, in the cases of L^2 subcritical and critical power nonlinearity. In particular, in the critical case, a key role is played by a thorough analysis of the Gagliardo-Nirenberg inequalities on metric graphs and by estimates of their best constants. Most of the techniques are new and suited to the investigation of variational problems on metric graphs.