We investigate the existence of ground states for the focusing nonlinear Schrödinger
equation on a prototypical doubly periodic metric graph. When the nonlinearity
power is below 4, ground states exist for every value of the mass, while, for
every nonlinearity power between 4 (included) and 6 (excluded), a mark of
-criticality
arises, as ground states exist if and only if the mass exceeds a threshold value that
depends on the power. This phenomenon can be interpreted as a continuous
transition from a two-dimensional regime, for which the only critical power is 4, to a
one-dimensional behavior, in which criticality corresponds to the power 6.
We show that such a dimensional crossover is rooted in the coexistence of
one-dimensional and two-dimensional Sobolev inequalities, leading to a new family of
Gagliardo–Nirenberg inequalities that account for this continuum of critical
exponents.