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arXiv 1401.2666math.AP
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欧氏半空间上临界非线性椭圆型方程组解的分类

Classification of Solutions to a Critically Nonlinear System of Elliptic Equations on Euclidean Half-Space

Mathew R. Gluck, Lei Zhang

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17 pages, 1 figure
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摘要：

对于$N\geq 3$和非负实数$a_{ij}$和$b_{ij}$（$i,j= 1, \cdots, m$），考虑半线性椭圆系统\begin{equation*} \begin{cases} Δu_i + \prod_{j = 1}^m u_j^{a_{ij}} = 0 & \text{ in } \mathbb R_+^N, \frac{\partial u_i}{\partial y_N} = c_i \prod_{j = 1}^m u_j^{b_{ij}} & \text{ on } \partial \mathbb R_+^N \end{cases} \qquad i = 1, \cdots, m \end{equation*} %，其中$\mathbb R_+^N$是$N$维欧氏空间的上半部分.在对指数$a_{ij}$和$b_{ij}$进行适当假设的情况下，证明了该系统正$C^2(\mathbb R_+^N)\cap C^1(\overline{\mathbb R_+^N})$ -解的分类定理。

英文摘要：

For $N\geq 3$ and non-negative real numbers $a_{ij}$ and $b_{ij}$ ($i,j= 1, \cdots, m$), the semi-linear elliptic system \begin{equation*} \begin{cases} Δu_i + \prod_{j = 1}^m u_j^{a_{ij}} = 0 & \text{ in } \mathbb R_+^N, \frac{\partial u_i}{\partial y_N} = c_i \prod_{j = 1}^m u_j^{b_{ij}} & \text{ on } \partial \mathbb R_+^N \end{cases} \qquad i = 1, \cdots, m \end{equation*} % is considered, where $\mathbb R_+^N$ is the upper half of $N$-dimensional Euclidean space. Under suitable assumptions on the exponents $a_{ij}$ and $b_{ij}$, a classification theorem for the positive $C^2(\mathbb R_+^N)\cap C^1(\overline{\mathbb R_+^N})$-solutions of this system is proven.

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