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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 |
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Vol. 12 (2019), No. 6, 1597–1612
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Abstract |
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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 -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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Keywords
metric graphs, Sobolev inequalities, threshold phenomena,
nonlinear Schrödinger equation
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Mathematical Subject Classification 2010
Primary: 35Q55, 35R02
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Milestones
Received: 7 May 2018
Revised: 7 September 2018
Accepted: 25 October 2018
Published: 7 February 2019
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Authors |
| Riccardo Adami | |
| Dipartimento di Scienze Matematiche
“G. L. Lagrange” Politecnico di Torino Torino Italy |
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| Simone Dovetta | |
| Dipartimento di Scienze Matematiche
“G. L. Lagrange” Politecnico di Torino Torino Italy |
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| Enrico Serra | |
| Dipartimento di Scienze Matematiche
“G. L. Lagrange” Politecnico di Torino Torino Italy |
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| Paolo Tilli | |
| Dipartimento di Scienze Matematiche
“G. L. Lagrange” Politecnico di Torino Torino Italy |
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