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\limits_{j=1}^m u_j^{a_{ij}}=0,\text{in}\mathbb{R}_+^N,\\dfrac{\partial u_i}{\partial y_N}=c_i\prod\limits_{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 $C^2(\mathbb{R}_+^N)\cap C^1(\overline{R_+^N})$-solutions of this system is proven.