Classification of Solutions to a Critically Nonlinear System of Elliptic   Equations on Euclidean Half-Space
  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}   \Delta u_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. 
Mathew R. Gluck, Lei Zhang
math.AP