Method of sub-super solutions for fractional elliptic equations

Yanqin Fang; De Tang

doi:10.3934/dcdsb.2017212

Article Contents

2018, Volume 23, Issue 8: 3153-3165. Doi: 10.3934/dcdsb.2017212

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Method of sub-super solutions for fractional elliptic equations

Yanqin Fang^1,2, and
De Tang^3, ,

1.
School of Mathematics, Hunan University, Changsha 410082, Hunan, China
2.
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, 2522, NSW, Australia
3.
School of Mathematics(Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong, China

* Corresponding author: tangdehnu@126.com
* Corresponding author: tangdehnu@126.com

Received: March 31, 2017

Revised: July 31, 2017

Early access: August 2017

Published: August 2018

The first author is supported by National Natural Sciences Foundations of China 11301166 and Young Teachers Program of Hunan University
The second author is supported by Natural Science Foundation of Hunan Province, China 2016JJ2018

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Abstract

By applying the method of sub-super solutions, we obtain the existence of weak solutions to fractional Laplacian

$\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{s}}u=f(x,u),&\text{in}\ \Omega , \\ u=0,&\text{in}\ {{\mathbb{R}}^{N}}\backslash \Omega , \\\end{array} \right.$

where $f:\Omega \text{ }\!\!\times\!\!\text{ }\mathbb{R}\to \mathbb{R}$ is a Caratheódory function.

Let $ν$ be a Radon measure. Based on the existence result in (1), we derive the existence of weak solutions for the semilinear fractional elliptic equation with measure data

$ \left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{s}}u=f(x,u)+\nu ,&\text{in}\ \Omega , \\ u=0,&\text{in}\ {{\mathbb{R}}^{N}}\backslash \Omega , \\\end{array} \right. $

Some results in[7] are extended.

In addition, we generalize some results to systems of fractional Laplacian equations by constructing subsolutions and supersolutions.

Keywords:

Mathematics Subject Classification: Primary: 35J60, 35J67; Secondary: 58J05.

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