Regularity and existence of positive solutions for a fractional system

<p style='text-indent:20px;'>We consider the nonlinear fractional elliptic system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{array}{ll} (- \Delta)^{\frac{\alpha_1}{2}}u(x) = f(x, u, v), &amp; \text{in}\, \, \, \Omega, \\ (- \Delta)^{\frac{\alpha_2}{2}}v(x) = g(x, u, v), &amp; \text{in}\, \, \, \Omega, \\ u = v = 0, &amp; \text{in}\, \, \, \mathbb{R}^n\setminus\Omega, \end{array} \right. \label{a-1.2} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;\alpha_1, \alpha_2&lt;2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain with <inline-formula><tex-math id="M3">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> boundary in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. To overcome the technical difficulty due to the different fractional orders, we employ two distinct methods and derive the a priori estimates for <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;\alpha_1, \alpha_2&lt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ 1&lt;\alpha_1, \alpha_2 &lt;2 $\end{document}</tex-math></inline-formula> respectively. Moreover, combining the a priori estimate with the topological degree theory, we prove the existence of positive solutions.</p>