Symmetry and monotonicity of integral equation systems

In this paper, we investigate positive solutions of the following integral equation system {u(x)=∫Rn|x−y|α−nua(y)vb(y)dy,v(x)=∫Rn|x−y|β−n|uc(y)vd(y)dy where a,b,c,d,α,βa,b,c,d,α,β are constants. The symmetry and monotonicity of its solutions are proved by the method of moving planes, the non-existence and the exact form are given by the method of moving spheres.