Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations

Silvia Cingolani, Louis Jeanjean, Kazunaga Tanaka

Research output: Contribution to journalArticle

2 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 ⊂ 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.

Original languageEnglish
Pages (from-to)37-66
Number of pages30
JournalJournal of Fixed Point Theory and Applications
Volume19
Issue number1
DOIs
Publication statusPublished - 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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