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

Abstract

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)^{α/2} u(x)=f(v(x)), & (-\lap)^{β/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 $g$.

References in corpus (1)

• Symmetry of Solutions to Semilinear Equations Involving the Fractional Laplacian on ${\mathbb{R}}^n$ and ${\mathbb{R}}_{+}^n$