Ground states for the NLS equation with combined nonlinearity on periodic metric graphs

We investigate the existence of ground states with prescribed mass for the Non-Linear Schr\"odinger energy with combined nonlinearities on $1$ and $2$-periodic metric graphs. This is the natural prosecution of previous studies concerning on the one hand the homogeneous NLS equation on periodic graphs, and on the other hand the NLS equation with combined nonlinearity on noncompact metric graphs with finitely many vertexes and edges. As in the latter case, it turns out that the interplay between different nonlinearities creates new phenomena with respect to the homogenous setting, but, due to the periodicity, in a quite different way; in particular, for $2$-periodic graphs, the so called dimensional crossover occurs. As a by-product, we extend existing results for the homogeneous NLS on the square and honeycomb grids to general $2$-periodic graphs. Furthermore, we also improve previous results obtained for the inhomogeneous NLS on noncompact graphs with finitely many vertexes and edges.