Ground states of the $L^2$-critical NLS equation with localized nonlinearity on a tadpole graph

TL;DR

This paper investigates the existence of ground states for the $L^2$-critical nonlinear Schrödinger equation on a tadpole graph, highlighting unique features of localized nonlinearity in this simplified model.

Contribution

It provides initial insights into ground state existence for the $L^2$-critical NLS on metric graphs with localized nonlinearity, focusing on the tadpole graph as a model.

Findings

Identifies conditions for ground state existence and nonexistence

Highlights specific features of localized nonlinearity on tadpole graphs

Provides a foundation for future research on nonlinear Schrödinger equations on graphs

Abstract

The paper aims at giving a first insight on the existence/nonexistence of ground states for the $L^2$-critical NLS equation on metric graphs with localized nonlinearity. In particular, we focus on the tadpole graph, which, albeit being a toy model, allows to point out some specific features of the problem, whose understanding will be useful for future investigations.