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

Norihisa Ikoma, Kazunaga Tanaka

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41 Citations (Scopus)


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
Issue number3
Publication statusPublished - 2011 Jan 1


ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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