### Abstract

For k = 1, · · ·, K, let M_{k} be a q_{k}-dimensional smooth compact framed manifold in ℝ^{N} with q_{k} ∈ {1, · · ·, N - 1}. We consider the equation -ε^{2}Δu + V(x)u - u^{p} = 0 in ℝ^{N} where for each k ∈ {1, · · ·, K} and some m_{k} > 0, V(x) = |dist(x, M_{k})|^{m}
_{k} + O(|dist(x, M_{k})|^{m}
_{k}+1) as dist(x, M_{k}) → 0. For a sequence of ε converging to zero, we will find a positive solution u_{ε} of the equation which concentrates on M_{1} ∪ · · · ∪ M_{K}.

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

Pages (from-to) | 825-842 |

Number of pages | 18 |

Journal | Communications on Pure and Applied Analysis |

Volume | 14 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 1 2015 |

### Fingerprint

### Keywords

- Concentration phenomena
- Infinite dimensional reduction
- Nondegeneracy
- Nonlinear schrdinger equation

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Analysis*,

*14*(3), 825-842. https://doi.org/10.3934/cpaa.2015.14.825

**Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations.** / Byeon, Jaeyoung; Kwon, Ohsang; Oshita, Yoshihito.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Analysis*, vol. 14, no. 3, pp. 825-842. https://doi.org/10.3934/cpaa.2015.14.825

}

TY - JOUR

T1 - Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations

AU - Byeon, Jaeyoung

AU - Kwon, Ohsang

AU - Oshita, Yoshihito

PY - 2015/5/1

Y1 - 2015/5/1

N2 - For k = 1, · · ·, K, let Mk be a qk-dimensional smooth compact framed manifold in ℝN with qk ∈ {1, · · ·, N - 1}. We consider the equation -ε2Δu + V(x)u - up = 0 in ℝN where for each k ∈ {1, · · ·, K} and some mk > 0, V(x) = |dist(x, Mk)|m k + O(|dist(x, Mk)|m k+1) as dist(x, Mk) → 0. For a sequence of ε converging to zero, we will find a positive solution uε of the equation which concentrates on M1 ∪ · · · ∪ MK.

AB - For k = 1, · · ·, K, let Mk be a qk-dimensional smooth compact framed manifold in ℝN with qk ∈ {1, · · ·, N - 1}. We consider the equation -ε2Δu + V(x)u - up = 0 in ℝN where for each k ∈ {1, · · ·, K} and some mk > 0, V(x) = |dist(x, Mk)|m k + O(|dist(x, Mk)|m k+1) as dist(x, Mk) → 0. For a sequence of ε converging to zero, we will find a positive solution uε of the equation which concentrates on M1 ∪ · · · ∪ MK.

KW - Concentration phenomena

KW - Infinite dimensional reduction

KW - Nondegeneracy

KW - Nonlinear schrdinger equation

UR - http://www.scopus.com/inward/record.url?scp=84923972774&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923972774&partnerID=8YFLogxK

U2 - 10.3934/cpaa.2015.14.825

DO - 10.3934/cpaa.2015.14.825

M3 - Article

AN - SCOPUS:84923972774

VL - 14

SP - 825

EP - 842

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

SN - 1534-0392

IS - 3

ER -