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

 Recent Issues Vol. 292: 1 Vol. 291: 1  2 Vol. 290: 1  2 Vol. 289: 1  2 Vol. 288: 1  2 Vol. 287: 1  2 Vol. 286: 1  2 Vol. 285: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Special Issues Submission Guidelines Submission Form Contacts Author Index To Appear ISSN: 0030-8730
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