Positive solutions for a weighted fractional system

In this article, we study positive solutions to the system

$\{\begin{array}{l}{A}_{\alpha }u\left(x\right)=C_{n,\alpha }PV{\int }_{R^n}\frac{a_1\left(x-y\right)\left(u\left(x\right)-u\left(y\right)\right)}{{\left|x-y\right|}^{n+\alpha }}\text{d}y=f\left(u\left(x\right),\upsilon \left(x\right)\right),\hfill \\ B_{\beta }\upsilon \left(x\right)=C_{n,\beta }PV{\int }_{R^n}\frac{a_2\left(x-y\right)\left(\upsilon \left(x\right)-\upsilon \left(y\right)\right)}{{\left|x-y\right|}^{n+\beta }}\text{d}y=g\left(u\left(x\right),\upsilon \left(x\right)\right).\hfill \end{array}$

To reach our aim, by using the method of moving planes, we prove a narrow region principle and a {decay at infinity} by the iteration method. On the basis of these results, we conclude radial symmetry and monotonicity of positive solutions for the problems involving the weighted fractional system on an unit ball and the whole space. Furthermore, non-existence of nonnegative solutions on a half space is given.