Qualitative analysis for an elliptic system in the punctured space

In this paper, we investigate the qualitative properties of positive solutions for the following two-coupled elliptic system in the punctured space: $$\{\begin{array}{l}-\mathrm{\Delta }u={\mu }_1{u}^{2q+1}+\beta u^q{v}^{q+1}\\ -\mathrm{\Delta }v={\mu }_2{v}^{2q+1}+\beta v^q{u}^{q+1}\end{array}\text{in }{\mathbb{R}}^n\mathrm{\backslash }\{0\},$$ where ${\mu }_1,{\mu }_2$ and $\beta$ are all positive constants, $n\ge 3$. 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.