  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.
