Multiple sign-changing and semi-nodal solutions for coupled Schrodinger equations
Multiple sign-changing and semi-nodal solutions for coupled Schrodinger equations
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DOI：
10.1016/j.jde.2013.08.009
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发表时间：
2013-04
期刊：
影响因子：
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通讯作者：
Zhijie Chen;Changshou Lin;W. Zou
中科院分区：
文献类型：
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作者：
Zhijie Chen;Changshou Lin;W. Zou
研究了在数学物理中出现的几种耦合薛定谔方程：{− Δ u1 + λ 1u 1= μ 1u 13 + β u1 u22，x∈ Ω，− Δ u2 + λ 2u 2= μ 2u 23 + β u12 u2，x∈ Ω，u1 = u2 = 0.这里Ω <$RN（N= 2，3）是光滑有界区域，λ 1，λ 2，μ 1，μ 2都是正常数.我们证明了，对于每个k∈ N，存在β k> 0，使得对于每个固定的β∈（0，β k），该系统至少有k个变号解（即两个分量都变号）和k个半结点解（即一个分量变号，另一个分量为正）.
We study the following coupled Schrödinger equations which have appeared as several models from mathematical physics:{− Δ u 1+ λ 1 u 1= μ 1 u 1 3+ β u 1 u 2 2, x∈ Ω,− Δ u 2+ λ 2 u 2= μ 2 u 2 3+ β u 1 2 u 2, x∈ Ω, u 1= u 2= 0 on∂ Ω. Here Ω⊂ R N (N= 2, 3) is a smooth bounded domain, λ 1, λ 2, μ 1, μ 2 are all positive constants. We show that, for each k∈ N there exists β k> 0 such that this system has at least k sign-changing solutions (ie, both two components change sign) and k semi-nodal solutions (ie, one component changes sign and the other one is positive) for each fixed β∈(0, β k).