Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains

In this paper, we first establish a narrow region principle for systems involving the fractional Laplacian in unbounded domains, which plays an important role carrying on direct method of moving planes. Then combining with sliding method, derive monotonicity bounded positive solutions to following Lipschitz domains \begin{document}$ \Omega $\end{document} style='text-indent:20px;'> \begin{document}$ \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ }ll} \left(-\Delta\right)^{s}u& = &f(u,v), & \mbox{in}\ \Omega\,, \\[0.05cm] \left(-\Delta\right)^{t}v& &g(u,v),& u,\,v&\equiv&0, \mbox{on}\ \mathbb{R}^{n}\setminus\Omega\,, \end{array}\right. \end{equation*} $\end{document} style='text-indent:20px;'>without any decay assumptions solution pair id="M2">\begin{document}$ (u,\,v) at infinity.