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

TL;DR

This paper establishes Liouville theorems for a class of fractional semilinear systems in the upper half space, using a direct moving planes method without decay assumptions, thus extending the understanding of nonnegative solutions.

Contribution

It introduces a novel application of the moving planes method to fractional systems in \\mathbb{R}^n_+ without requiring decay at infinity, broadening Liouville theorem applicability.

Findings

Liouville theorems for fractional systems in upper half space

Method applicable without decay assumptions at infinity

Conditions on nonlinearities f and g ensure nonexistence of solutions

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 $g$.