  In the paper, we consider the fractional elliptic system
\begin{equation*}\left\{\begin{array}{ll} (-
\Delta)^{\frac{\alpha_1}{2}}u(x)+\sum\limits^n_{i=1}b_i(x)\frac{\partial
u}{\partial x_i}+B(x)u(x)=f(x,u,v),& \mbox { in } \Omega,\\ (-
\Delta)^{\frac{\alpha_2}{2}}v(x)+\sum\limits^n_{i=1}c_i(x)\frac{\partial
v}{\partial x_i}+C(x)v(x)=g(x,u,v),& \mbox { in } \Omega,\\ u=v=0, & \mbox { in
} \mathbb{R}^n\setminus\Omega, \end{array} \right.\label{a-1.2} \end{equation*}
where $\Omega$ is a bounded domain with $C^2$ boundary in $\mathbb{R}^n$ and
$n>\max\{\alpha_1,\alpha_2\}$. We first utilize the blowing-up and re-scaling
method to derive the a priori estimate for positive solutions when
$1<\alpha_1,\alpha_2 <2$. Then for $0<\alpha_1,\alpha_2 <1$, we obtain the
regularity estimate of positive solutions. On top of this, using the
topological degree theory we prove the existence of positive solutions.
