Existence and symmetry of positive solutions of an integral equation system

In this paper, we investigate positive solutions of the following integral equation system in R n : { u ( x ) = ∫ R n ∣ x − y ∣ α − n v ( y ) p d y , v ( x ) = ∫ R n ∣ x − y ∣ β − n u ( y ) q d y , where p , q > 1 , 0 < α , β < n . With the method of moving spheres, we show the existence and the exact form of its solution in the case p ≤ ( n + α ) / ( n − β ) , q ≤ ( n + β ) / ( n − α ) ; and with the method of moving planes, we prove the symmetry and monotonicity of its solution in the case 1 p + 1 + 1 q + 1 = n − α 2 n + β − α + n − β 2 n + α − β .