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

Abstract

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 Schrödinger 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.

22 pages, 7 figures. Keywords: metric graphs, NLS, ground states, localized nonlinearity, $L^2$-critical case. Some minor revisions have been made with respect to the previous version. Accepted for publication by Calc. Var. Partial Differential Equations