Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^n_+ $

Mei Yu; Xia Zhang; Binlin Zhang

doi:10.3934/cpaa.2020157

Article Contents

2020, Volume 19, Issue 7: 3597-3612. Doi: 10.3934/cpaa.2020157

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Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^n_+ $

1.
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an, 710129, China
2.
Department of Mathematics, Yeshiva University, New York, 10033, USA
3.
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China
4.
College of Mathematics and Systems Sciences, Shandong University of Science and Technology, Qingdao, 266590, China

^*Corresponding author
^*Corresponding author

Received: May 31, 2019

Revised: December 31, 2019

Published: April 2020

M. Yu was supported by National Natural Science Foundation of China (Grant No. 11801446, Grant No. 11971385), Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2018JQ1037, Grant No. 2019JM-275); X. Zhang was supported by National Natural Science Foundation of China (Grant No. 11671111, 11871177); B. Zhang was supported by National Natural Science Foundation of China (Grant No. 11871199), Heilongjiang Province Postdoctoral Startup Foundation (LBH-Q18109), and the Introduction and Cultivation Project of Young and Innovative Talents in Universities of Shandong Province

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Abstract

In this paper, we consider the following equation with the higher-order fractional Laplacian $ (-\Delta)^s $ for $ s = m+\frac{\alpha}{2} $:

$ \begin{equation*} (-\Delta)^{s} u(x) = f(u(x)), \qquad x\in\mathbb{R}^n_+, \end{equation*} $

where $ m\in \mathbb{N}^* $, $ 0<\alpha<2 $. By developing a narrow region principle in unbounded domain and establishing a equivalence of differential equation and integral equation, together with the method of moving planes, we deduce the monotonicity property of positive solutions and the Liouville theorem of nonnegative solutions.

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Mathematics Subject Classification: Primary: 35S15, 35B06; Secondary: 35J61.

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