Sala Seminari (Dip. Matematica).
We investigate the existence of ground states for a focusing NLS functional with a power nonlinearity over a metric graph (i.e. a one-dimensional network made up of finite intervals and/or half-lines, glued together according to the topology of a graph). The admissible functions are subject to a mass constraint, and we will discuss how the value of the prescribed mass, the topology of the graph and its metric properties may interact, in order to guarantee or prevent existence of minimisers. Finally, we will also consider some cases where the power nonlinearity is critical.