# American Institute of Mathematical Sciences

October  2018, 23(8): 3153-3165. doi: 10.3934/dcdsb.2017212

## Method of sub-super solutions for fractional elliptic equations

 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

Received  April 2017 Revised  August 2017 Published  September 2017

Fund Project: 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

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.
Citation: Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212
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