Negative Energy Ground States for the L 2-Critical NLSE on Metric Graphs

Abstract

We investigate the existence of ground states with prescribed mass for the focusing nonlinear Schrödinger equation with L 2 -critical power nonlinearity on noncompact quantum graphs. We prove that, unlike the case of the real line, for certain classes of graphs there exist ground states with negative energy for a whole interval of masses.A key role is played by a thorough analysis of Gagliardo-Nirenberg inequalities and on estimates of the optimal constants. Most of the techniques are new and suited to the investigation of variational problems on metric graphs.

Variational and Stability Properties of Constant Solutions to the NLS Equation on Compact Metric Graphs

“…Postponing the details to Section 3, we recall that it has been recently proved in [19] that in the critical regime a ground state exists if and only if the mass does not exceed a threshold value depending on the topology of the graph. More precisely, this threshold equals the value µ R + , the critical mass on the half-line, if G has at least a terminal edge (i.e., an edge ending into a vertex of degree 1, see Figure 2), whereas it equals µ R , the critical mass on the real line, if there is no edge of this kind (see for instance [7] and Section 3 here for further details on µ R + and µ R ).…”

“…The existence of mass-constrained ground states was initiated in the series of works [1,2,3,4], for the special case of a star-graph (a graph obtained by merging N halflines). For generic non-compact metric graphs the problem was addressed in [5,6] in the subcritical regime and in [7] in the critical one (see also [30]). We also note the paper [17], where the stationary solutions for the NLS equation on the tadpole graph (a loop with one half-line attached to it) were characterized and where the existence of the ground state was only conjectured (the conjecture was then confirmed in [6]).…”

Quantum graphs and dimensional crossover: the honeycomb

“…Starting from this negative result, the problem of ensuring (or excluding) the existence of ground states for the NLS on graphs gained some popularity in the community, and some general results were found, isolating a key topological condition ( [5]), studying in detail particular cases ( [14,26,27]), dealing with compact graphs ( [12,13]), introducing concentrated nonlinearities ( [15,28,29,30]), focusing on the more challenging L 2 -critical case (i.e. p = 6 [6]). More recently, also some pioneering investigations of nonlinear Dirac equations has been initiated ( [9,10]).…”

An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity

“…On the other hand, in the case α = 0 which is usually called Kirchhoff Laplacian (and which will be denoted simply by −∆ G in place of −∆ δ,0 G in the sequel, for the sake of simplicity) more general graphs have been studied (precisely, any graph satisfying (H1)-(H2)). We mention, in this regard, [7,8,9] for a discussion of the existence of ground states (i.e., those standing waves that minimize the energy functional associated with (2)) and [11,21,36,41,42] concerning, more generally, excited states. We also mention [22] where the same problems are studied in the presence of an external potential.…”