Regularity of solutions for a system of integral equations

$u(x) = \int_{R^{n}} |x-y|^{\alpha -n} v(y)^q dy$, $ v(x) = \int_{R^{n}} |x-y|^{\alpha -n} u(y)^p dy$

with $\frac{1}{q+1}+\frac{1}{p+1}=\frac{n-\alpha}{n}$. In our previous paper, under the natural integrability conditions $u \in L^{p+1} (R^n)$ and $v \in L^{q+1} (R^n)$, we prove that all the solutions are radially symmetric and monotone decreasing about some point. In this paper, we go further to study the regularity of the solutions. We show that the solutions are bounded, and hence continuous and smooth. We also prove that if $p = q$, then $u = v$, and they both must assume the standard form

$ c(\frac{t}{t^2 + |x - x_o|^2})^{(n-\alpha)/2} $

with some constant $c = c(n, \alpha)$, and for some $t > 0$ and $x_o \in R^n$.