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

DOI: 10.2140/pjm.2017.290.301

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:

$$\left\{\begin{array}{c}\hfill \\ \hfill \\ {\left(-\Delta \right)}^{\alpha ∕2}u\left(x\right)=v^q\left(x\right),\hfill & x\in {ℝ}^n,\hfill \\ \hfill \\ \hfill \\ {\left(-\Delta \right)}^{\alpha ∕2}v\left(x\right)=u^p\left(x\right),\hfill & x\in {ℝ}^n.\hfill \\ \hfill \\ \hfill \\ \hfill \end{array}\right.$$

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

Received: 29 November 2015

Revised: 13 October 2016

Accepted: 14 December 2016

Published: 25 July 2017