Regularity and existence of positive solutions for a fractional system

We consider the nonlinear fractional elliptic system\begin{document}$ \begin{equation*} \left\{\begin{array}{ll} (- \Delta)^{\frac{\alpha_1}{2}}u(x) = f(x, u, v), & \text{in}\, \, \, \Omega, \\ (- \Delta)^{\frac{\alpha_2}{2}}v(x) = g(x, u, v), & \text{in}\, \, \, \Omega, \\ u = v = 0, & \text{in}\, \, \, \mathbb{R}^n\setminus\Omega, \end{array} \right. \label{a-1.2} \end{equation*} $\end{document}where \begin{document}$ 0<\alpha_1, \alpha_2<2 $\end{document} and \begin{document}$ \Omega $\end{document} is a bounded domain with \begin{document}$ C^2 $\end{document} boundary in \begin{document}$ \mathbb{R}^n $\end{document} . To overcome the technical difficulty due to the different fractional orders, we employ two distinct methods and derive the a priori estimates for \begin{document}$ 0<\alpha_1, \alpha_2<1 $\end{document} and \begin{document}$ 1<\alpha_1, \alpha_2 <2 $\end{document} respectively. Moreover, combining the a priori estimate with the topological degree theory, we prove the existence of positive solutions.