Optimal number of solutions for nonlinear coupled Schrodinger systems, part I: Synchronized case
Optimal number of solutions for nonlinear coupled Schrodinger systems, part I: Synchronized case
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非线性耦合薛定谔系统的最优解数，第一部分：同步情况
DOI：
10.1016/j.jde.2018.09.018
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发表时间：
2019
影响因子：
2.4
通讯作者：
Wang Lushun
中科院分区：
文献类型：
--
作者：
Tang Zhongwei;Wang Lushun
本文考虑如下非线性薛定谔系统(Aε){−ε2Δu+u=μ1 u 3+βu v 2 inΩ，−ε2Δv+v=μ2 v 3+βu 2 v inΩ，u>；0，v>；0 inΩ，∂u∂ν=∂v∂ν=0 on∂Ω，其中ε>；0，μ1>；0，μ2>；0，β∈(0，min{μ1，μ2})∪(max{μ1，μ2}，+∞)，Ω是R3中边界光滑的有界域，ν是定义在∂Ω上的外单位法线，Ω的边界。通过李雅普诺夫-施密特约化证明，存在ε0>；0使得对每个0<；ε<；ε0和每个满足1≤k≤δ(Ω)ε3的整数k，(Aε)有一个具有k个内部尖峰的同步解，其中δ(Ω)是一个仅依赖于Ω的常数.此外，k的上界是最优的，且(Aε)恰好有O(1/ε3)多个同步解。
In this paper, we consider the following nonlinear Schrödinger systems (A ε){− ε 2 Δ u+ u= μ 1 u 3+ β u v 2 in Ω,− ε 2 Δ v+ v= μ 2 v 3+ β u 2 v in Ω, u> 0, v> 0 in Ω,∂ u∂ ν=∂ v∂ ν= 0 on∂ Ω, where ε> 0, μ 1> 0, μ 2> 0, β∈(0, min{μ 1, μ 2})∪(max{μ 1, μ 2},+∞), Ω is a bounded domain with smooth boundary in R 3, and ν is the outward unit normal defined on∂ Ω, the boundary of Ω. By Lyapunov–Schmidt reduction argument, we prove that there exists ε 0> 0 such that for each 0< ε< ε 0 and each integer k satisfying 1≤ k≤ δ (Ω) ε 3,(A ε) has a synchronized solution with k interior spikes, where δ (Ω) is a constant depending only on Ω. Moreover, the upper bound of k is optimal and (A ε) has exactly O (1/ε 3) many synchronized solutions.