Calculus of Variations and Geometric Measure Theory

S. Dovetta - L. Tentarelli

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

created by tentarelli on 07 Jul 2022


Published Paper

Inserted: 7 jul 2022
Last Updated: 7 jul 2022

Journal: Calc. Var. Partial Differential Equations
Volume: 58
Number: 3
Pages: art. 108, 26pp
Year: 2019
Doi: 10.1007/s00526-019-1565-5

ArXiv: 1811.02387 PDF


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.