We establish a direct method of moving planes for systems of fractional Laplacian
equations. By using this direct method of moving planes, we obtain symmetry and
nonexistence of positive solutions for the following system of fractional Laplacian
equations:

the fractional Laplacian, maximum principles for
antisymmetric functions, narrow region principle, decay at
infinity, method of moving planes, radial symmetry,
nonexistence of positive solutions