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

Norihisa Ikoma, Kazunaga Tanaka

    研究成果: Article

    40 引用 (Scopus)

    抄録

    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.

    元の言語English
    ページ(範囲)449-480
    ページ数32
    ジャーナルCalculus of Variations and Partial Differential Equations
    40
    発行部数3
    DOI
    出版物ステータスPublished - 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

    これを引用

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    abstract = "We consider a singular perturbation problem for a system of nonlinear Schr{\"o}dinger 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.",
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