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Method of sub-super solutions for fractional elliptic equations

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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  • 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.

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


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