## 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

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.apde/1552356130

Digital Object Identifier
doi:10.2140/apde.2019.12.1597

Mathematical Reviews number (MathSciNet)
MR3921313

#### Citation

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. https://projecteuclid.org/euclid.apde/1552356130

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