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September  2017, 16(5): 1707-1718. doi: 10.3934/cpaa.2017082

## A direct method of moving planes for a fully nonlinear nonlocal system

 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaan xi, 710129, China

*The Corresponding author

Received  September 2016 Revised  January 2017 Published  May 2017

In this paper we consider the system involving fully nonlinear nonlocal operators:
 $\left\{\begin{array}{ll}{\mathcal F}_{α}(u(x)) = C_{n,α} PV ∈t_{\mathbb{R}^n} \frac{F(u(x)-u(y))}{|x-y|^{n+α}} dy=v^p(x)+k_1(x)u^r(x),\\{\mathcal G}_{β}(v(x)) = C_{n,β} PV ∈t_{\mathbb{R}^n} \frac{G(v(x)-v(y))}{|x-y|^{n+β}} dy=u^q(x)+k_2(x)v^s(x),\end{array}\right.$
where
 $0<α, β<2,$
 $p, q, r, s>1,$
 $k_1(x), k_2(x)\geq0.$
A narrow region principle and a decay at infinity are established for carrying on the method of moving planes. Then we prove the radial symmetry and monotonicity for positive solutions to the nonlinear system in the whole space. Furthermore non-existence of positive solutions to the system on a half space is derived.
Citation: Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082
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