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2019 Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs
Riccardo Adami, Simone Dovetta, Enrico Serra, Paolo Tilli
Anal. PDE 12(6): 1597-1612 (2019). DOI: 10.2140/apde.2019.12.1597

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.

Citation

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Riccardo Adami. Simone Dovetta. Enrico Serra. Paolo Tilli. "Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs." Anal. PDE 12 (6) 1597 - 1612, 2019. https://doi.org/10.2140/apde.2019.12.1597

Information

Received: 7 May 2018; Revised: 7 September 2018; Accepted: 25 October 2018; Published: 2019
First available in Project Euclid: 12 March 2019

zbMATH: 07061134
MathSciNet: MR3921313
Digital Object Identifier: 10.2140/apde.2019.12.1597

Subjects:
Primary: 35Q55, 35R02

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.12 • No. 6 • 2019
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