Solutions of fully nonlinear nonlocal systems

In this paper we consider the system involving fully nonlinear nonlocal operators:{Fα(u(x))=Cn,αPV∫RnG(u(x)−u(y))|x−y|n+αdy=f(v(x)),Fβ(v(x))=Cn,βPV∫RnG(v(x)−v(y))|x−y|n+βdy=g(u(x)). For carrying on the method of moving planes, a narrow region principle and a decay at infinity are established. 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 nonlinear system on a half space is derived.