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We investigate the existence of ground states for the focusing nonlinearSchroedinger equation on a prototypical doubly periodic metric graph. When thenonlinearity power is below 4, ground states exist for every value of the mass,while, for every nonlinearity power between 4 (included) and 6 (excluded), amark of $L^2$-criticality arises, as ground states exist if and only if themass exceeds a threshold value that depends on the power. This phenomenon canbe interpreted as a continuous transition from a two-dimensional regime, forwhich the only critical power is 4, to a one-dimensional behavior, in whichcriticality corresponds to the power 6. We show that such a dimensionalcrossover is rooted in the coexistence of one-dimensional and two-dimensionalSobolev inequalities, leading to a new family of Gagliardo-Nirenberginequalities that account for this continuum of critical exponents.

Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs