Symmetry properties in systems of fractional Laplacian equations

Zhigang Wu; Hao Xu

doi:10.3934/dcds.2019068

Article Contents

2019, Volume 39, Issue 3: 1559-1571. Doi: 10.3934/dcds.2019068

This issue Previous Article Liouville's theorem for a fractional elliptic system Next Article Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations

Symmetry properties in systems of fractional Laplacian equations

Zhigang Wu^1, and
Hao Xu^2, ,

1.
Department of Applied Mathematics, Donghua University, Shanghai 201620, China
2.
Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO 80302, USA

* Corresponding author: Hao Xu
* Corresponding author: Hao Xu

Received: March 31, 2018

Revised: April 30, 2018

Published: December 2018

The first author is supported by Natural Science Foundation of Shanghai grant 16ZR1402100

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We consider the systems of fractional Laplacian equations in a domain(bounded or unbounded) in $\mathbb{R}^n$. By using a direct method of moving planes, we show that $u_i(x)$ ($i = 1,2,···,m$) are radial symmetric about the same point and strictly decreasing in the radial direction with respect to this point. Comparing with Zhuo-Chen-Cui-Yuan [38], our results not only include subcritical case and critical case but also include supercritical case, and we need not the nonlinear terms to be homogenous. In addition, we completely remove the nonnegativity of $\frac{\partial f_i}{\partial u_i}$. Above all, to the best of our knowledge, it is the first result on the symmetric property of the system containing the gradient of the solution in the nonlinear terms.

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Mathematics Subject Classification: Primary: 35B50; Secondary: 35R11.

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