RESEARCH ARTICLE

A radial symmetry and Liouville theorem for systems involving fractional Laplacian

  • Dongsheng LI ,
  • Zhenjie LI
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  • School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Received date: 02 Aug 2015

Accepted date: 18 Jan 2016

Published date: 20 Feb 2017

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

We investigate the nonnegative solutions of the system involving the fractional Laplacian:

{(Δ)αui(x)=fi(u),xn,i=1,2,...,m,u(x)=(u1(x),u2(x),...,um(x)),

Where 0α1, n2, fi(u),1≤im , are real-valued nonnegative functions of homogeneous degree pi≥0 and nondecreasing with respect to the independent variables u1, u2, . . . , um. By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x0 if pi=(n+2α)/(n2α) for each 1≤im; and the only nonnegative solution of this system is u ≡ 0 if 1pi(n+2α)/(n2α) for all 1≤im.

Cite this article

Dongsheng LI , Zhenjie LI . A radial symmetry and Liouville theorem for systems involving fractional Laplacian[J]. Frontiers of Mathematics in China, 2017 , 12(2) : 389 -402 . DOI: 10.1007/s11464-016-0517-z

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