Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations

Silvia Cingolani; Louis Jeanjean; Kazunaga Tanaka

doi:10.1007/s11784-016-0347-3

Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations

Silvia Cingolani, Louis Jeanjean, Kazunaga Tanaka

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations LA,Vħu=f(|u|2)uinRNwhere N≥ 3 , LA,Vħ is the Schrödinger operator with a magnetic field having source in a C¹ vector potential A and a scalar continuous (electric) potential V defined by LA,Vħ=-ħ2Δ-2ħiA·∇+|A|2-ħidivA+V(x).Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain Ω ⊂ R^N such that (Formula presented.). For ħ> 0 small we prove the existence of at least cupl (K) + 1 geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as ħ→ 0.

Original language	English
Pages (from-to)	37-66
Number of pages	30
Journal	Journal of Fixed Point Theory and Applications
Volume	19
Issue number	1
DOIs	https://doi.org/10.1007/s11784-016-0347-3
Publication status	Published - 2017 Mar 1

Keywords

Complex-valued solutions
Cuplength
Magnetic fields
Nonlinear Schrödinger equations
Semiclassical limit

ASJC Scopus subject areas

Modelling and Simulation
Geometry and Topology
Applied Mathematics

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title = "Multiple complex-valued solutions for nonlinear magnetic Schr{\"o}dinger equations",

abstract = "We study, in the semiclassical limit, the singularly perturbed nonlinear Schr{\"o}dinger equations LA,Vħu=f(|u|2)uinRNwhere N≥ 3 , LA,Vħ is the Schr{\"o}dinger operator with a magnetic field having source in a C1 vector potential A and a scalar continuous (electric) potential V defined by LA,Vħ=-ħ2Δ-2ħiA·∇+|A|2-ħidivA+V(x).Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain Ω ⊂ RN such that (Formula presented.). For ħ> 0 small we prove the existence of at least cupl (K) + 1 geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as ħ→ 0.",

keywords = "Complex-valued solutions, Cuplength, Magnetic fields, Nonlinear Schr{\"o}dinger equations, Semiclassical limit",

author = "Silvia Cingolani and Louis Jeanjean and Kazunaga Tanaka",

year = "2017",

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doi = "10.1007/s11784-016-0347-3",

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TY - JOUR

T1 - Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations

AU - Cingolani, Silvia

AU - Jeanjean, Louis

AU - Tanaka, Kazunaga

PY - 2017/3/1

Y1 - 2017/3/1

N2 - We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations LA,Vħu=f(|u|2)uinRNwhere N≥ 3 , LA,Vħ is the Schrödinger operator with a magnetic field having source in a C1 vector potential A and a scalar continuous (electric) potential V defined by LA,Vħ=-ħ2Δ-2ħiA·∇+|A|2-ħidivA+V(x).Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain Ω ⊂ RN such that (Formula presented.). For ħ> 0 small we prove the existence of at least cupl (K) + 1 geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as ħ→ 0.

AB - We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations LA,Vħu=f(|u|2)uinRNwhere N≥ 3 , LA,Vħ is the Schrödinger operator with a magnetic field having source in a C1 vector potential A and a scalar continuous (electric) potential V defined by LA,Vħ=-ħ2Δ-2ħiA·∇+|A|2-ħidivA+V(x).Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain Ω ⊂ RN such that (Formula presented.). For ħ> 0 small we prove the existence of at least cupl (K) + 1 geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as ħ→ 0.

KW - Complex-valued solutions

KW - Cuplength

KW - Magnetic fields

KW - Nonlinear Schrödinger equations

KW - Semiclassical limit

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