Regularity of solutions for a system of integral equations

Abstract

In this paper, we study positive solutions of the following system of integral equations in $R^n$: <br> <p align="center"> $u(x) = \int_{R^{n}} |x-y|^{\alpha -n} v(y)^q dy$, $ v(x) = \int_{R^{n}} |x-y|^{\alpha -n} u(y)^p dy$ <p align="left" class="times"> 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 <br> <p align="center"> $ c(\frac{t}{t^2 + |x - x_o|^2})^{(n-\alpha)/2} $ <p align="left" class="times">with some constant $c = c(n, \alpha)$, and for some $t > 0$ and $x_o \in R^n$.