A system of integral equations on half space

Let R + n be the n-dimensional upper half Euclidean space, and let α be any real number satisfying 0 < α < n , we study positive solutions of the following system of integral equations in R + n : { u ( x ) = ∫ R + n ( 1 | x − y | n − α − 1 | x ⁎ − y | n − α ) v q ( y ) d y ; v ( x ) = ∫ R + n ( 1 | x − y | n − α − 1 | x ⁎ − y | n − α ) u p ( y ) d y where x ⁎ is the reflection of the point x about the plane x n = 0 . We assume that v ∈ L q + 1 ( R + n ) , u ∈ L p + 1 ( R + n ) with 1 q + 1 + 1 p + 1 = n − α n . In our previous paper, we considered the corresponding single equation u ( x ) = ∫ R + n ( 1 | x − y | n − α − 1 | x ⁎ − y | n − α ) u n + α n − α ( y ) d y and proved that every positive solution of the above integral equation is rotationally symmetric about some line parallel to x n -axis. We also established regularity of solutions. Now we go further to study the regularity and rotational symmetry for solutions of the above system of integral equations and generalize the results in our previous paper.