 1862 Riccardo Adami, Simone Dovetta, Enrico Serra, Paolo Tilli
 Dimensional
crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs
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May 7, 18

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Abstract. We investigate the existence of ground states for the focusing nonlinear Schroedinger 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 L^2criticality 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 twodimensional regime, for which the only critical power is 4, to a onedimensional behavior, in which criticality corresponds to the power $6$. We show that such a dimensional crossover is rooted in the coexistence of onedimensional and twodimensional Sobolev inequalities, leading to a new family of GagliardoNirenberg inequalities that account for this continuum of critical exponents.
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