Abstract
In this paper, we consider the following non-linear system involving the fractional Laplacian
0.1
\begin{equation} \left\{\begin{array}{@{}ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. \end{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
$x_n$
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.
Publisher
Cambridge University Press (CUP)