  In this paper, we investigate the qualitative properties of positive
solutions for the following two-coupled elliptic system in the punctured space:
$$ \begin{cases} -\Delta u =\mu_1 u^{2q+1} + \beta u^q v^{q+1} \\ -\Delta v
=\mu_2 v^{2q+1} + \beta v^q u^{q+1} \end{cases} \textmd{in} ~\mathbb{R}^n
\backslash \{0\}, $$ where $\mu_1, \mu_2$ and $\beta$ are all positive
constants, $n\geq 3$. We establish a monotonicity formula that completely
characterizes the singularity of positive solutions. We prove a sharp global
estimate for both components of positive solutions. We also prove the
nonexistence of positive semi-singular solutions, which means that one
component is bounded near the singularity and the other component is unbounded
near the singularity.
