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arXiv · 1611.09133

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

Lizhi Zhang·Mei Yu·Jianming He
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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)^{\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 $g$.

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Lizhi Zhang, Mei Yu, Jianming He. 2016-11-28. A Liouville Theorem for a Class of Fractional Systems in $\mathbb{R}^n_+$. https://arxiv.org/abs/1611.09133

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