A generalised Nehari manifold method for a class of non-linear Schrödinger systems in ℝ3

Tommaso Cortopassi; Vladimir Georgiev

doi:10.1063/5.0084041

A generalised Nehari manifold method for a class of non-linear Schrödinger systems in ℝ³

Tommaso Cortopassi^*, Vladimir Georgiev

^*Corresponding author for this work

Graduate School of Advanced Science and Engineering

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

We study the existence of positive solutions of a particular elliptic system in ℝ3 composed of two non linear stationary Schrödinger equations (NLSEs), that is -∈2Δu + V(x)u = hv(u, v), -∈2Δv + V(x)v = hu(u, v). Under certain hypotheses on the potential V and the non linearity h, we manage to prove that there exists a solution (u∈, v∈) that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as ∈ → 0. We also estimate this energy in terms of particular ground state energies. This work follows closely what is done in [6], although here we consider a more general non linearity and we restrict ourselves to the case where the domain is ℝ3.

Original language	English
Title of host publication	8th International Conference New Trends in the Applications of Differential Equations in Sciences, NTADES 2021
Editors	Angela Slavova
Publisher	American Institute of Physics Inc.
ISBN (Electronic)	9780735441866
DOIs	https://doi.org/10.1063/5.0084041
Publication status	Published - 2022 Apr 5
Event	8th International Conference New Trends in the Applications of Differential Equations in Sciences, NTADES 2021 - St. Constantin and Helena, Bulgaria Duration: 2021 Sept 6 → 2021 Sept 10

Publication series

Name	AIP Conference Proceedings
Volume	2459
ISSN (Print)	0094-243X
ISSN (Electronic)	1551-7616

Conference

Conference	8th International Conference New Trends in the Applications of Differential Equations in Sciences, NTADES 2021
Country/Territory	Bulgaria
City	St. Constantin and Helena
Period	21/9/6 → 21/9/10

ASJC Scopus subject areas

Physics and Astronomy(all)

Access to Document

10.1063/5.0084041

Cite this

Cortopassi, T., & Georgiev, V. (2022). A generalised Nehari manifold method for a class of non-linear Schrödinger systems in ℝ³. In A. Slavova (Ed.), 8th International Conference New Trends in the Applications of Differential Equations in Sciences, NTADES 2021 [030003] (AIP Conference Proceedings; Vol. 2459). American Institute of Physics Inc.. https://doi.org/10.1063/5.0084041

A generalised Nehari manifold method for a class of non-linear Schrödinger systems in ℝ³. / Cortopassi, Tommaso; Georgiev, Vladimir.

8th International Conference New Trends in the Applications of Differential Equations in Sciences, NTADES 2021. ed. / Angela Slavova. American Institute of Physics Inc., 2022. 030003 (AIP Conference Proceedings; Vol. 2459).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Cortopassi, T & Georgiev, V 2022, A generalised Nehari manifold method for a class of non-linear Schrödinger systems in ℝ³. in A Slavova (ed.), 8th International Conference New Trends in the Applications of Differential Equations in Sciences, NTADES 2021., 030003, AIP Conference Proceedings, vol. 2459, American Institute of Physics Inc., 8th International Conference New Trends in the Applications of Differential Equations in Sciences, NTADES 2021, St. Constantin and Helena, Bulgaria, 21/9/6. https://doi.org/10.1063/5.0084041

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abstract = "We study the existence of positive solutions of a particular elliptic system in ℝ3 composed of two non linear stationary Schr{\"o}dinger equations (NLSEs), that is -∈2Δu + V(x)u = hv(u, v), -∈2Δv + V(x)v = hu(u, v). Under certain hypotheses on the potential V and the non linearity h, we manage to prove that there exists a solution (u∈, v∈) that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as ∈ → 0. We also estimate this energy in terms of particular ground state energies. This work follows closely what is done in [6], although here we consider a more general non linearity and we restrict ourselves to the case where the domain is ℝ3. ",

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N2 - We study the existence of positive solutions of a particular elliptic system in ℝ3 composed of two non linear stationary Schrödinger equations (NLSEs), that is -∈2Δu + V(x)u = hv(u, v), -∈2Δv + V(x)v = hu(u, v). Under certain hypotheses on the potential V and the non linearity h, we manage to prove that there exists a solution (u∈, v∈) that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as ∈ → 0. We also estimate this energy in terms of particular ground state energies. This work follows closely what is done in [6], although here we consider a more general non linearity and we restrict ourselves to the case where the domain is ℝ3.

AB - We study the existence of positive solutions of a particular elliptic system in ℝ3 composed of two non linear stationary Schrödinger equations (NLSEs), that is -∈2Δu + V(x)u = hv(u, v), -∈2Δv + V(x)v = hu(u, v). Under certain hypotheses on the potential V and the non linearity h, we manage to prove that there exists a solution (u∈, v∈) that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as ∈ → 0. We also estimate this energy in terms of particular ground state energies. This work follows closely what is done in [6], although here we consider a more general non linearity and we restrict ourselves to the case where the domain is ℝ3.

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