The properties of positive solutions to semilinear equations involving the fractional Laplacian

Rongrong Yang; Zhongxue Lü

doi:10.3934/cpaa.2019052

Article Contents

2019, Volume 18, Issue 3: 1073-1089. Doi: 10.3934/cpaa.2019052

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The properties of positive solutions to semilinear equations involving the fractional Laplacian

Rongrong Yang and
Zhongxue Lü^,

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

* Corresponding author
* Corresponding author

Received: November 30, 2017

Revised: March 31, 2018

Published: May 2019

The second author is supported by NSFC(No.11271166), NSF of Jiangsu Province(No. BK2010172), sponsored by Qing Lan Project

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Abstract

Let $Ω$ be either a unit ball or a half space. Consider the following Dirichlet problem involving the fractional Laplacian

$\left\{ \begin{array}{*{35}{l}} \begin{align} & {{(-\Delta )}^{\frac{\alpha }{2}}}u=f(u),\ \ \text{in}\ \ \Omega , \\ & u=0, ~~~~~~~~~~~~~~~~~~~~ \text{in}\ \ {{\Omega }^{c}},\ \\ \end{align} & \ & {} \\\end{array} \right.~~~~(1)$

where $α$ is any real number between $zhongwenzy$ and $$. Under some conditions on $f$, we study the equivalent integral equation

$ \begin{align}u(x) \ = \ \int{{}}_{ Ω}G(x, y)f(u(y))dy, \end{align}~~~~(2) $

here $G(x, y)$ is the Green's function associated with the fractional Laplacian in the domain $Ω$. We apply the method of moving planes in integral forms to investigate the radial symmetry, monotonicity and regularity for positive solutions in the unit ball. Liouville type theorems-non-existence of positive solutions in the half space are also deduced.

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Mathematics Subject Classification: Primary: 35J60, 35B06, 35B09, 35B53, 45G10, 45M20.

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