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

Abstract

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.

arXiv comment21 pages, 3 figures

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Cited by in corpus (6)

• Standing waves on quantum graphs
• Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs
• Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover
• Some remarks on a minimization problem associated to a fourth order nonlinear Schrödinger equation
• An existence theory for nonlinear equations on metric graphs via energy methods
• Non-Kirchhoff Vertices and NLS Ground States on graphs