Bound states of the NLS equation on metric graphs with localized nonlinearities

Abstract

We investigate the existence of multiple bound states of prescribed mass for the nonlinear Schrödinger equation on a noncompact metric graph. The main feature is that the nonlinearity is localized only in a compact part of the graph. Our main result states that for every integer k, the equation possesses at least k solutions of prescribed mass, provided that the mass is large enough. These solutions arise as constrained critical points of the NLS energy functional. Estimates for the energy of the solutions are also established.

Existence of infinitely many stationary solutions of the L2-subcritical and critical NLSE on compact metric graphs

“…As proved in [21] (Section 2), since {u n } n∈N is a bounded Palais-Smale sequence, condition (ii) in Definition 2.2 can be conveniently rewritten as J(u n ) → 0 in H −1 (G). Moreover, if λ n := λ(u n ) is as in (11), then {λ n } n∈N is bounded in R, and (up to subsequences)…”

“…Moreover, existence of ground states and bound states on non-compact graphs was investigated also for the NLS equation with concentrated nonlinearity in [23,22,21]. Specifically, the general scheme followed in [21] provides the tools we will use in this paper when dealing with bound states (see Section 2).…”

Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit

“…Moreover, we point out that Theorem 2.13, in contrast to Theorem 2.12, holds only for a fixed range of power exponents, namely the so-called L 2 -subcritical case p ∈ (2, 6). However, this is the only range of powers for which multiplicity results are known for the NLSE (see [51]). On the other hand, these results are parametrized by the L 2 norm of the wave function while Theorem 2.13 is parametrized by the frequency and hence (in some sense) it presents as a byproduct a new result for the NLSE.…”

“…Following [31,41], also a simplified version of this model has recently gained a particular attention: the case of a nonlinearity localized on the compact core K of the graph (which is the subgraph consisting of all the bounded edges); namely, − u ′′ − χ K |u| p−2 u = λu (2) with Kirchhoff vertex conditions and χ K denoting the characteristic function of K. This problem has been studied in the L 2 -subcritical case in [51,52,54], while some new results on the L 2 -critical case have been presented in [24,25] (for a general overview see also [16]). Remark 1.1.…”

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

“…The strategy used in [47], in order to solve these problems is, then, the following: -detect the energy levels at which the Palais-Smale condition is satisfied, namely detect the values c ∈ R such that any sequence (u n ) ∈ H µ (G) satisfying…”