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 L2-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.

Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs / Adami, Riccardo; Dovetta, Simone; Serra, Enrico; Tilli, Paolo. - In: ANALYSIS & PDE. - ISSN 1948-206X. - STAMPA. - 12:6(2019), pp. 1597-1612. [10.2140/apde.2019.12.1597]

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

Adami, Riccardo;Dovetta, Simone;Serra, Enrico;Tilli, Paolo
2019

Abstract

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 L2-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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2729937