Ground States of the L 2-Critical NLS Equation with Localized Nonlinearity on a Tadpole Graph

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. As a consequence, 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. More precisely, we prove that there exists an interval of masses for which ground states do exist, and that for large masses the functional is unbounded from below, whereas for small masses ground states cannot exist although the functional is bounded.

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

“…As mentioned in the Introduction, the key Gagliardo-Nirenberg inequality of the problem with localized nonlinearity is not the standard one given by ( 14), but instead (12) where the L 6 term affects only the compact core of the graph. In particular, it is evident by (11) that the value of the best constant C K is the crucial parameter in order to determine whether solutions of (2) exist or not.…”

“…Problem (2) was first proposed in [12], in the "toy" model of the tadpole graph (see Figure 1). Here we aim at extending that seminal result in order to present an (almost) complete classification of the whole phenomenology.…”

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Carrying on the discussion initiated in [12], 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 [4].

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NLS ground states on metric trees: existence results and open questions

“…Within this framework, there has been an intensive study of the existence of mass‐constrained ground states for the NLS energy, that is, global minimizers of the energy among functions of prescribed norm. This problem has been initially considered in the case of graphs made up of a core of finitely many bounded edges, and a finite number of unbounded edges (half lines) attached to it, and this setting is nowadays quite well understood (we refer to [5–7] for the nonlinearity extended to the whole graph, and to [18, 19, 30, 31, 34] for the nonlinearity concentrated on the sole compact core). Similar results have then been accomplished also in the case of compact graphs [13, 16, 24].…”

Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one

“…Since then, the problem has been studied both on the real line R (see [4,5] and [9,10] for the rigorous derivation from the standard NLS equation) and in higher dimensions (see [2,3] for the three dimensional case and [1,12,13] for the problem in dimension two). More recently, a similar setting has been considered also on non-compact metric graphs (see [18,19,29,30,33] and [6,7] for the nonlinear Dirac equation).…”