Symmetry properties in systems of fractional Laplacian equations

We consider the systems of fractional Laplacian equations in a domain(bounded or unbounded) in $\mathbb{R}^n$. By using a direct method of moving planes, we show that $u_i(x)$ ($i = 1,2,···,m$) are radial symmetric about the same point and strictly decreasing in the radial direction with respect to this point. Comparing with Zhuo-Chen-Cui-Yuan [38], our results not only include subcritical case and critical case but also include supercritical case, and we need not the nonlinear terms to be homogenous. In addition, we completely remove the nonnegativity of $\frac{\partial f_i}{\partial u_i}$. Above all, to the best of our knowledge, it is the first result on the symmetric property of the system containing the gradient of the solution in the nonlinear terms.