NLS ground states on metric graphs with localized nonlinearities

Tentarelli, Lorenzo

doi:10.1016/j.jmaa.2015.07.065

Cited by 47 publications

(60 citation statements)

References 21 publications

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“…Both are well known when G = R + , and by a rearrangement argument they are proved to be valid on any noncompact metric graph G, with the same constant as on R + (see [4,18] for more details). In the following we will need to compare the achievable energy levels on G with the ground-state energy levels on the real line R and on the halfline R + , exploiting the fact that in the prototypical cases where G = R or G = R + everything is known explicitly.…”

Section: Some Basic Tools (Old and New)mentioning

confidence: 97%

Multiple positive bound states for the subcritical NLS equation on metric graphs

2018

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We consider the Schrödinger equation with a subcritical focusing power nonlinearity on a noncompact metric graph, and prove that for every finite edge there exists a threshold value of the mass, beyond which there exists a positive bound state achieving its maximum on that edge only. This bound state is characterized as a minimizer of the energy functional associated to the NLS equation, with an additional constraint (besides the mass prescription): this requires particular care in proving that the minimizer satisfies the Euler?Lagrange equation. As a consequence, for a sufficiently large mass every finite edge of the graph hosts at least one positive bound state that, owing to its minimality property, is orbitally stable.

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Section: Some Basic Tools (Old and New)mentioning

confidence: 97%

Multiple positive bound states for the subcritical NLS equation on metric graphs

2018

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“…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).…”

Section: Introductionmentioning

confidence: 99%

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

Dovetta¹

2018

Journal of Differential Equations

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Borrelli¹,

Carlone²,

Tentarelli³

2019

SIAM J. Math. Anal.

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In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the L 2 -subcritical case, they converge to the bound states of the NLS equation in the nonrelativistic limit. 1The attention recently attracted by the linear and the nonlinear Dirac equations is due to their applications, as effective equations, in many physical models, as in solid state physics and nonlinear optics [33,34].While originally the NLDE appeared as a field equation for relativistic interacting fermions [38], then it was used in particle physics to simulate features of quark confinement, acoustic physics, and in the context of Bose-Einstein condensates [34].Recently, it also appeared that some properties of physical models, as thin carbon structures, are well described using as an effective equation for non-relativistic electronic properties , the Dirac equation. We mention, thereupon, the seminal papers by Fefferman and Weinstein [28,29], the work of Arbunich and Sparber [10] (where a rigorous justification of linear and nonlinear equations in two-dimensional honeycomb structures is given) and the referenced therein. In addition, we

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NLS ground states on metric graphs with localized nonlinearities

Cited by 47 publications

References 21 publications

Multiple positive bound states for the subcritical NLS equation on metric graphs

Multiple positive bound states for the subcritical NLS equation on metric graphs

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

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

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