L2 -critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features

Carrying on the discussion initiated in Dovetta and Tentarelli (Ground states of the L^2-critical NLS equation with localized nonlinearity on a tadpole graph, 2018. arXiv:1804.11107 [math.AP]), we investigate the existence of ground states of prescribed mass for the L^2-critical NonLinear Schrödinger Equation 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 et al. (Commun Math Phys 352(1):387–406, 2017). Our results rely on a thorough analysis of the optimal constant of a suitable variant of the L 2 -critical Gagliardo–Nirenberg inequality.