NLS ground states on metric trees: existence results and open questions

We consider the minimization of the NLS energy on a metric tree, either rooted or unrooted, subject to a mass constraint. With respect to the same problem on other types of metric graphs, several new features appear, such as the existence of minimizers with positive energy, and the emergence of unex...

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Bibliographic Details
Published in:Journal of the London Mathematical Society 2020-12, Vol.102 (3), p.1223-1240
Main Authors: Dovetta, Simone, Serra, Enrico, Tilli, Paolo
Format: Article
Language:English
Subjects:
35Q55
35R02
49J40
58E30 (primary)
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Summary:We consider the minimization of the NLS energy on a metric tree, either rooted or unrooted, subject to a mass constraint. With respect to the same problem on other types of metric graphs, several new features appear, such as the existence of minimizers with positive energy, and the emergence of unexpected threshold phenomena. We also study the problem with a radial symmetry constraint that is in principle different from the free problem due to the failure of the Pólya–Szegő inequality for radial rearrangements. A key role is played by a new Poincaré inequality with remainder.
ISSN:0024-6107
1469-7750
DOI:10.1112/jlms.12361