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arXiv:1811.02387 [math.AP]AbstractReferencesReviewsResources

$L^2$-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features

Simone Dovetta, Lorenzo Tentarelli

Published 2018-11-06Version 1

Carrying on the discussion initiated in (Dovetta-Tentarelli'18), we investigate the existence of ground states of prescribed mass for the $L^2$-critical NonLinear Schr\"odinger Equation (NLSE) on noncompact metric graphs with localized nonlinearity. Precisely, we show that the existence (or nonexistence) of ground states mainly depends on a parameter called reduced critical mass, and then we discuss how the topological and metric features of the graphs affect such a parameter, establishing some relevant differences with respect to the case of the extended nonlinearity studied by (Adami-Serra-Tilli'17). Our results rely on a thorough analysis of the optimal constant of a suitable variant of the $L^2$-critical Gagliardo-Nirenberg inequality.

Comments: 22 pages, 7 figures. Keywords: metric graphs, NLS, ground states, localized nonlinearity, $L^2$-critical case
Categories: math.AP, math-ph, math.FA, math.MP
Subjects: 35R02, 35Q55, 81Q35, 35Q40, 49J40
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