Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations

Silvia Cingolani, Louis Jeanjean, Kazunaga Tanaka

    Research output: Contribution to journalArticle

    1 Citation (Scopus)

    Abstract

    We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations (Formula presented.)where (Formula presented.), (Formula presented.) is the Schrödinger operator with a magnetic field having source in a (Formula presented.) vector potential A and a scalar continuous (electric) potential V defined by (Formula presented.)Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain (Formula presented.) such that (Formula presented.)and we set (Formula presented.). For (Formula presented.) small we prove the existence of at least (Formula presented.) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as (Formula presented.).

    Original languageEnglish
    Pages (from-to)1-30
    Number of pages30
    JournalJournal of Fixed Point Theory and Applications
    DOIs
    Publication statusAccepted/In press - 2016 Nov 14

    Fingerprint

    Nonlinear equations
    Magnetic fields
    Electric potential
    Semiclassical Limit
    Vector Potential
    Electric Potential
    Singularly Perturbed
    Bounded Domain
    Modulus
    Nonlinear Equations
    Magnetic Field
    Scalar
    Distinct

    Keywords

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

    ASJC Scopus subject areas

    • Modelling and Simulation
    • Geometry and Topology
    • Applied Mathematics

    Cite this

    Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations. / Cingolani, Silvia; Jeanjean, Louis; Tanaka, Kazunaga.

    In: Journal of Fixed Point Theory and Applications, 14.11.2016, p. 1-30.

    Research output: Contribution to journalArticle

    @article{c5c26f5a42da43b1af91b14e8cb53791,
    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 (Formula presented.)where (Formula presented.), (Formula presented.) is the Schr{\"o}dinger operator with a magnetic field having source in a (Formula presented.) vector potential A and a scalar continuous (electric) potential V defined by (Formula presented.)Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain (Formula presented.) such that (Formula presented.)and we set (Formula presented.). For (Formula presented.) small we prove the existence of at least (Formula presented.) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as (Formula presented.).",
    keywords = "Complex-valued solutions, Cuplength, Magnetic fields, Nonlinear Schr{\"o}dinger equations, Semiclassical limit",
    author = "Silvia Cingolani and Louis Jeanjean and Kazunaga Tanaka",
    year = "2016",
    month = "11",
    day = "14",
    doi = "10.1007/s11784-016-0347-3",
    language = "English",
    pages = "1--30",
    journal = "Journal of Fixed Point Theory and Applications",
    issn = "1661-7738",
    publisher = "Springer Science + Business Media",

    }

    TY - JOUR

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

    AU - Cingolani, Silvia

    AU - Jeanjean, Louis

    AU - Tanaka, Kazunaga

    PY - 2016/11/14

    Y1 - 2016/11/14

    N2 - We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations (Formula presented.)where (Formula presented.), (Formula presented.) is the Schrödinger operator with a magnetic field having source in a (Formula presented.) vector potential A and a scalar continuous (electric) potential V defined by (Formula presented.)Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain (Formula presented.) such that (Formula presented.)and we set (Formula presented.). For (Formula presented.) small we prove the existence of at least (Formula presented.) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as (Formula presented.).

    AB - We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations (Formula presented.)where (Formula presented.), (Formula presented.) is the Schrödinger operator with a magnetic field having source in a (Formula presented.) vector potential A and a scalar continuous (electric) potential V defined by (Formula presented.)Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain (Formula presented.) such that (Formula presented.)and we set (Formula presented.). For (Formula presented.) small we prove the existence of at least (Formula presented.) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as (Formula presented.).

    KW - Complex-valued solutions

    KW - Cuplength

    KW - Magnetic fields

    KW - Nonlinear Schrödinger equations

    KW - Semiclassical limit

    UR - http://www.scopus.com/inward/record.url?scp=84995414745&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=84995414745&partnerID=8YFLogxK

    U2 - 10.1007/s11784-016-0347-3

    DO - 10.1007/s11784-016-0347-3

    M3 - Article

    SP - 1

    EP - 30

    JO - Journal of Fixed Point Theory and Applications

    JF - Journal of Fixed Point Theory and Applications

    SN - 1661-7738

    ER -