### Abstract

We consider a singular perturbation problem for a system of nonlinear Schrödinger equations:where N = 2, 3, μ_{1}, μ_{2}, β > 0 and V_{1}(x), V_{2}(x): R^{N} → (0, ∞) are positive continuous functions. We consider the case where the interaction β > 0 is relatively small and we define for P ε R^{N} 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 ⊂ R^{N} 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 P_{0} ε Λ with m(P_{0}) = inf_{PεΛ}m(P). Moreover (v_{1e{open}}(x), v_{2e{open}}(x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.

Original language | English |
---|---|

Pages (from-to) | 449-480 |

Number of pages | 32 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 40 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2011 Jan 1 |

Externally published | Yes |

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### 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.

Research output: Contribution to journal › Article

*Calculus of Variations and Partial Differential Equations*, vol. 40, no. 3, pp. 449-480. https://doi.org/10.1007/s00526-010-0347-x

}

TY - JOUR

T1 - A local mountain pass type result for a system of nonlinear Schrödinger equations

AU - Ikoma, Norihisa

AU - Tanaka, Kazunaga

PY - 2011/1/1

Y1 - 2011/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=78751581933&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78751581933&partnerID=8YFLogxK

U2 - 10.1007/s00526-010-0347-x

DO - 10.1007/s00526-010-0347-x

M3 - Article

AN - SCOPUS:78751581933

VL - 40

SP - 449

EP - 480

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 3

ER -