March  2005, 4(1): 1-8. doi: 10.3934/cpaa.2005.4.1

Regularity of solutions for a system of integral equations

1. 

Department of Mathematics, Yeshiva University, New York, NY 10033, United States

2. 

Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0524

Received  April 2004 Revised  October 2004 Published  December 2004

In this paper, we study positive solutions of the following system of integral equations in $R^n$:

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

Citation: Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure & Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1
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