Analysis & PDE

  • Anal. PDE
  • Volume 12, Number 6 (2019), 1597-1612.

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

Riccardo Adami, Simone Dovetta, Enrico Serra, and Paolo Tilli

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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.

Article information

Anal. PDE, Volume 12, Number 6 (2019), 1597-1612.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces)

metric graphs Sobolev inequalities threshold phenomena nonlinear Schrödinger equation


Adami, Riccardo; Dovetta, Simone; Serra, Enrico; Tilli, Paolo. Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs. Anal. PDE 12 (2019), no. 6, 1597--1612. doi:10.2140/apde.2019.12.1597.

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