Vol. 12, No. 6, 2019

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

Riccardo Adami, Simone Dovetta, Enrico Serra and Paolo Tilli

Vol. 12 (2019), No. 6, 1597–1612
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 ${L}^{2}$-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.

Keywords
metric graphs, Sobolev inequalities, threshold phenomena, nonlinear Schrödinger equation
Mathematical Subject Classification 2010
Primary: 35Q55, 35R02