#### Vol. 290, No. 2, 2017

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A direct method of moving planes for the system of the fractional Laplacian

### Chunxia Cheng, Zhongxue Lü and Yingshu Lü

Vol. 290 (2017), No. 2, 301–320
##### Abstract

We establish a direct method of moving planes for systems of fractional Laplacian equations. By using this direct method of moving planes, we obtain symmetry and nonexistence of positive solutions for the following system of fractional Laplacian equations:

 $\left\{\begin{array}{c}\phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ {\left(-\Delta \right)}^{\alpha ∕2}u\left(x\right)={v}^{q}\left(x\right),\phantom{\rule{1em}{0ex}}\hfill & x\in {ℝ}^{n},\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ {\left(-\Delta \right)}^{\alpha ∕2}v\left(x\right)={u}^{p}\left(x\right),\phantom{\rule{1em}{0ex}}\hfill & x\in {ℝ}^{n}.\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \end{array}\right\$
##### Keywords
the fractional Laplacian, maximum principles for antisymmetric functions, narrow region principle, decay at infinity, method of moving planes, radial symmetry, nonexistence of positive solutions
##### Mathematical Subject Classification 2010
Primary: 35B09, 35B50, 35B53, 35J61
##### Milestones
Revised: 13 October 2016
Accepted: 14 December 2016
Published: 25 July 2017
##### Authors
 Chunxia Cheng School of Mathematics and Statistics Jiangsu Normal University Xuzhou, 221116 China Zhongxue Lü School of Mathematics and Statistics Jiangsu Normal University Xuzhou, 221116 China Yingshu Lü Department of Mathematics Shanghai Jiao Tong University Shanghai, 200240 China