Positive Least Energy Solutions and Phase Separation for Coupled Schrödinger Equations with Critical Exponent
Positive Least Energy Solutions and Phase Separation for Coupled Schrödinger Equations with Critical Exponent
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DOI：
10.1007/s00205-012-0513-8
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发表时间：
2012-05
影响因子：
2.5
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
本文研究了以下耦合Schrödinger系统，它可以看作是一个临界耦合摄动Brezis-Nirenberg问题：$$\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.$$为光滑有界区域，且，其中为−Δ在Dirichlet边界条件下的第一特征值。请注意，非线性和耦合项在维度4中都是关键的（即当n = 4时）。我们证明了该临界系统对于负β、正小β和正大β具有正的最小能量解。在这种情况下，我们得到了正最小能量解的唯一性。我们还研究了在排斥情况下最小能量解的极限行为，并期望相分离。这些似乎是这个Schrödinger系统在临界情况下的第一个结果。
In this paper we study the following coupled Schrödinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: $$\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.$$Here,is a smooth bounded domain,and, whereis the first eigenvalue of −Δ with the Dirichlet boundary condition. Note that the nonlinearity and the coupling terms are both critical in dimension 4 (that is,whenN= 4). We show that this critical system has a positive least energy solution for negativeβ, positive smallβand positive largeβ. For the case in which, we obtain the uniqueness of positive least energy solutions. We also study the limit behavior of the least energy solutions in the repulsive case, and phase separation is expected. These seem to be the first results for this Schrödinger system in the critical case.