Qualitative properties of solutions for system involving the fractional Laplacian

In this paper, we consider the following non-linear system involving the fractional Laplacian 0. 1 equation \array{@{ll} (-) ^s u (x) = f (u, \, v), \\ (-) ^s v (x) = g (u, \, v), array. equation in two different types of domains, one is bounded, and the other is an infinite cylinder, where 0< s<1. We employ the direct sliding method for fractional Laplacian, different from the conventional extension and moving planes methods, to derive the monotonicity of solutions for (0. 1) in xₙ variable. Meanwhile, we develop a new iteration method for systems in the proofs. Hopefully, the iteration method can also be applied to solve other problems.