Radial symmetry results for fractional Laplacian systems

In this paper, we generalize the direct method of moving planes for the fractional Laplacian to the system case. Considering a coupled nonlinear system with fractional Laplacian, we first establish a decay at infinity principle and a narrow region principle. Using these principles, we obtain two radial symmetry results for the decaying solutions of the fractional Laplacian systems. Our method can be applied to fractional Schr\"odinger systems and fractional H\'enon systems.