A Liouville Theorem for a Class of Fractional Systems in ${\mathbb{R}}_{+}^n$

Let $0<\alpha ,\beta <2$ be any real number. In this paper, we investigate the following semilinear system involving the fractional Laplacian \begin{equation*} \left{\begin{array}{lll} (-\lap)^{\alpha/2} u(x)=f(v(x)), & (-\lap)^{\beta/2} v(x)=g(u(x)), & \qquad x\in$\mathbb{R}$^n_+, u,v\geq0, & \qquad x\in$\mathbb{R}$^n\setminus$\mathbb{R}$^n_+. \end{array}\right. \end{equation*} Applying a direct method of moving planes for the fractional Laplacian, without any decay assumption on the solutions at infinity, we prove Liouville theorems of nonnegative solutions under some natural conditions on $f$ and $g$.