Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case
Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case
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DOI：
10.1007/s00526-014-0717-x
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发表时间：
2015
影响因子：
2.1
通讯作者：
Zhijie Chen;W. Zou
中科院分区：
文献类型：
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作者：
Zhijie Chen;W. Zou
本文研究了与玻色-爱因斯坦凝聚有关的非线性薛定谔系统：Δ u+ λ 1u = μ 1u2 <$-1 + β u2 <$2 - 1v2 <$2，x∈ Ω，-Δ v+ λ 2v = μ 2v2 <$-1 + β v2 <$2 - 1u2 <$2，x∈ Ω，u≥ 0，v≥ 0在Ω上，u= v= 0在Ω上。其中Ω <$\mathbb R^ N Ω <$RN是一个光滑有界区域，2^*：= 2N N-2 2 <$：= 2 NN-2是Sobolev临界指数，-λ 1（Ω）<λ 1，λ 2 < 0-λ 1（Ω）< λ 1，λ 2< 0，μ 1，μ 2> 0 μ 1，μ 2> 0和β <$0 β <$0，其中λ 1（Ω）λ 1（Ω）是-Δ-Δ在Dirichlet边界条件下的第一特征值.当β= 0 β= 0时，这就是著名的Brezis-Nirenberg问题。作者在（Arch Ration Mech Anal 205：515-551，2012）中研究了N = 4的特殊情况。本文考虑高维情形N ≥ 5.有趣的是，我们可以证明对任意β <$0 β <$0（在N = 4的特殊情况下不成立），存在正的最小能量解（u β，v β）（u β，v β）.我们还研究了（u β，v β）（u β，v β）在β →-∞ β→-∞时的极限行为，并预测了相分离。特别地，当N ≥ 6时，u β-v β u β-v β将收敛于Brezis-Nirenberg问题的变号解.当λ _1= λ _2 λ_1 = λ_2时，还研究了最小能量解的分类问题。结果表明，与N= 4的特殊情况相比，出现了一些完全不同的现象。
We study the following nonlinear Schrödinger system which is related to Bose–Einstein condensate:{.-Δ u+ λ 1 u= μ 1 u 2∗-1+ β u 2∗ 2-1 v 2∗ 2, x∈ Ω,-Δ v+ λ 2 v= μ 2 v 2∗-1+ β v 2∗ 2-1 u 2∗ 2, x∈ Ω, u≥ 0, v≥ 0 in Ω, u= v= 0 on∂ Ω. Here Ω ⊂\mathbb R^ N Ω⊂ RN is a smooth bounded domain, 2^*:= 2N N-2 2∗:= 2 NN-2 is the Sobolev critical exponent,-λ _1 (Ω)< λ _1, λ _2< 0-λ 1 (Ω)< λ 1, λ 2< 0, μ _1, μ _2> 0 μ 1, μ 2> 0 and β ≠ 0 β≠ 0, where λ _1 (Ω) λ 1 (Ω) is the first eigenvalue of-Δ-Δ with the Dirichlet boundary condition. When β= 0 β= 0, this is just the well-known Brezis–Nirenberg problem. The special case N= 4 N= 4 was studied by the authors in (Arch Ration Mech Anal 205: 515–551, 2012). In this paper we consider the higher dimensional case N ≥ 5 N≥ 5. It is interesting that we can prove the existence of a positive least energy solution (u_ β, v_ β)(u β, v β) for any β ≠ 0 β≠ 0 (which can not hold in the special case N= 4 N= 4). We also study the limit behavior of (u_ β, v_ β)(u β, v β) as β →-∞ β→-∞ and phase separation is expected. In particular, u_ β-v_ β u β-v β will converge to sign-changing solutions of the Brezis–Nirenberg problem, provided N ≥ 6 N≥ 6. In case λ _1= λ _2 λ 1= λ 2, the classification of the least energy solutions is also studied. It turns out that some quite different phenomena appear comparing to the special case N= 4 N= 4.