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

Norihisa Ikoma, Kazunaga Tanaka

Research output: Contribution to journalArticle

39 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 Jan 1
Externally publishedYes

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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, 01.01.2011, p. 449-480.

Research output: Contribution to journalArticle

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