Number of synchronized and segregated interior spike solutions for nonlinear coupled elliptic systems

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• 作者：Zhongwei Tang ; ¹ ; ^{tangzw@bnu.edu.cn} ; Lushun Wang
• 关键词：Lyapunov&ndash ; Schmidt reduction methods ; Synchronized and segregated solution ; Interior spikes ; Elliptic systems
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 April 2017
• 年：2017
• 卷：448
• 期：2
• 页码：1079-1119
• 全文大小：626 K
• 卷排序：448

文摘

In this paper, we consider the following nonlinear coupled elliptic systemsequation(AεAε){−ε2Δu+u=μ1u3+βuv2in Ω,−ε2Δv+v=μ2v3+βu2vin Ω,u>0,v>0in Ω,∂u∂ν=∂v∂ν=0on ∂Ω, where ε>0ε>0, μ1>0μ1>0, μ2>0μ2>0, β∈Rβ∈R, and Ω is a bounded domain with smooth boundary in R3R3. Due to Lyapunov–Schmidt reduction method, we proved that (AεAε) has at least O(1ε3|ln⁡ε|) synchronized and segregated vector solutions for ε   small enough and some β∈Rβ∈R. Moreover, for each m∈(0,3)m∈(0,3) there exist synchronized and segregated vector solutions for (AεAε) with energies in the order of ε3−mε3−m. Our result extends the result of Lin, Ni and Wei [20], from the Lin–Ni–Takagi problem to the nonlinear elliptic systems.