  In this paper, we consider the following nonlinear system involving the
fractional Laplacian \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{s} u
(x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. (1)
\end{equation} in two different types of domains, one is bounded, and the other
is unbounded, where $0<s<1$. To investigate the qualitative properties of
solutions for fractional equations, the conventional methods are extension
method and moving planes method. However, the above methods have technical
limits in asymmetric and convex domains and so on. In this work, we employ the
direct sliding method for fractional Laplacian to derive the monotonicity of
solutions for (1) in $x_n$ variable in different types of domains. Meanwhile,
we develop a new iteration method for systems in the proofs which hopefully can
be applied to solve other problems.
