Liouville type theorems for a system of integral equations on upper half space

In this paper, we consider the following system of integral equations on upper half space .............u(x)=.Rn+(1|x.y|n...1|x..y|n..)(.1up1(y)+.1vp2(y)+.1 up3(y)vp4(y))dy;v(x)=.Rn+(1|x.y|n...1|x..y|n..)(.2uq1(y)+.2vq2(y)+.2uq3( y)vq4(y))dy, where . n+ = {x = (x 1, x 2, ..., x n ) . .n|x n > 0}, x.=(x1,x2,.,xn.1,.xn) is the reflection of the point x about the hyperplane x n = 0, 0 < . < n, . i , µ i , . i . 0 (i = 1, 2) are constants, p i . 0 and q i . 0 (i = 1, 2, 3, 4). We prove the nonexistence of positive solutions to the above system with critical and subcritical exponents via moving sphere method.