Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains

Lingwei Ma; Zhenqiu Zhang

doi:10.3934/dcds.2020268

Article Contents

2021, Volume 41, Issue 2: 537-552. Doi: 10.3934/dcds.2020268

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Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains

Lingwei Ma^1, and
Zhenqiu Zhang^2, ,

1.
School of Mathematical Sciences, Nankai University, Tianjin 300071, China
2.
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

^* Corresponding author: Zhenqiu Zhang
^* Corresponding author: Zhenqiu Zhang

Received: December 2019

Revised: May 2020

Early access: July 2020

Published: February 2021

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Abstract

In this paper, we first establish a narrow region principle for systems involving the fractional Laplacian in unbounded domains, which plays an important role in carrying on the direct method of moving planes. Then combining this direct method with the sliding method, we derive the monotonicity of bounded positive solutions to the following fractional Laplacian systems in unbounded Lipschitz domains $ \Omega $

$ \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} \left(-\Delta\right)^{s}u& = &f(u,v), & \mbox{in}\ \ \Omega\,, \\[0.05cm] \left(-\Delta\right)^{t}v& = &g(u,v),& \mbox{in}\ \ \Omega\,, \\[0.05cm] u,\,v&\equiv&0, & \mbox{on}\ \ \mathbb{R}^{n}\setminus\Omega\,, \end{array}\right. \end{equation*} $

without any decay assumptions on the solution pair $ (u,\,v) $ at infinity.

Keywords:

Fractional Laplacian systems,
unbounded Lipschitz domains,
narrow region principle in unbounded domains,
direct method of moving planes,
sliding method,
monotonicity.

Mathematics Subject Classification: Primary:35R11;Secondary:35B50, 35B99.

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