A local mountain pass type result for a system of nonlinear Schrödinger equations

Norihisa Ikoma, Kazunaga Tanaka

    Research output: Contribution to journalArticle

    38 Citations (Scopus)

    Abstract

    We consider a singular perturbation problem for a system of nonlinear Schrödinger equations:where N = 2, 3, μ1, μ2, β > 0 and V1(x), V2(x): RN → (0, ∞) are positive continuous functions. We consider the case where the interaction β > 0 is relatively small and we define for P ε RN the least energy level m(P) for non-trivial vector solutions of the rescaled "limit" problem: We assume that there exists an open bounded set Λ ⊂ RN satisfying We show that (*) possesses a family of non-trivial vector positive solutions which concentrates-after extracting a subsequence e{open}n → 0-to a point P0 ε Λ with m(P0) = infPεΛm(P). Moreover (v1e{open}(x), v2e{open}(x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.

    Original languageEnglish
    Pages (from-to)449-480
    Number of pages32
    JournalCalculus of Variations and Partial Differential Equations
    Volume40
    Issue number3
    DOIs
    Publication statusPublished - 2011

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    Mountain Pass
    System of Nonlinear Equations
    Nonlinear equations
    Singular Perturbation Problems
    Rescaling
    Bounded Set
    Subsequence
    Energy Levels
    Open set
    Electron energy levels
    Positive Solution
    Continuous Function
    Converge
    Energy
    Interaction

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    Cite this

    A local mountain pass type result for a system of nonlinear Schrödinger equations. / Ikoma, Norihisa; Tanaka, Kazunaga.

    In: Calculus of Variations and Partial Differential Equations, Vol. 40, No. 3, 2011, p. 449-480.

    Research output: Contribution to journalArticle

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