Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations

Jaeyoung Byeon; Ohsang Kwon; Yoshihito Oshita

doi:10.3934/cpaa.2015.14.825

Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations

Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

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 -ε²Δu + 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 M₁ ∪ · · · ∪ 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	https://doi.org/10.3934/cpaa.2015.14.825
Publication status	Published - May 1 2015

Keywords

Concentration phenomena
Infinite dimensional reduction
Nondegeneracy
Nonlinear schrdinger equation

ASJC Scopus subject areas

Analysis
Applied Mathematics

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title = "Standing wave concentrating on compact manifolds for nonlinear Schr{\"o}dinger equations",

abstract = "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)|mk + O(|dist(x, Mk)|mk+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.",

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author = "Jaeyoung Byeon and Ohsang Kwon and Yoshihito Oshita",

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AU - Byeon, Jaeyoung

AU - Kwon, Ohsang

AU - Oshita, Yoshihito

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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)|mk + O(|dist(x, Mk)|mk+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)|mk + O(|dist(x, Mk)|mk+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

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