In this paper, we consider the following equation with the higher-order fractional Laplacian $ (-\Delta)^s $ for $ s = m+\frac{\alpha}{2} $:
$ \begin{equation*} (-\Delta)^{s} u(x) = f(u(x)), \qquad x\in\mathbb{R}^n_+, \end{equation*} $
where $ m\in \mathbb{N}^* $, $ 0<\alpha<2 $. By developing a narrow region principle in unbounded domain and establishing a equivalence of differential equation and integral equation, together with the method of moving planes, we deduce the monotonicity property of positive solutions and the Liouville theorem of nonnegative solutions.
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