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

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2019,（3）
• 页码范围: 509-534
• 页数: 26页
• 语种: 英文

摘要

In this paper,we consider the following nonlinear coupled elliptic systems with continuous potentials:{-ε2△u + (1 + δP(x))u =μ1u3 + βuv2 in Ω,-ε2△v + (1 + δQ(x))v =μ2v3 + βu2v in Ω,u ＞ 0,v ＞ 0 in Ω,(e)u/(e)v=(e)v/(e)v=0 on (e)Ω,where Ω is a smooth bounded domain in RN for N =2,3,δ,ε,μ1 and μ2 are positive parameters,β ∈ R,P(x) and Q(x) are two smooth potentials defined on （Ω）,the closure of Ω.Due to Liapunov-Schmidt reduction method,we prove that (Aε) has at least O(1/(ε| lnε|)N) synchronized and O(1/(ε| lnε|)2N) segregated vector solutions for ε and δ small enough and some β ∈ R.Moreover,for each m ∈ (0,N) there exist synchronized and segregated vector solutions for (Aε) with energies in the order of εN-m.Our results extend the result of Lin et al.(2007) from the Lin-Ni-Takagi problem to the nonlinear Schr（o）dinger elliptic systems with continuous potentials.