IJPAM: Volume 52, No. 4 (2009)

POSITIVE REGULAR SOLUTIONS
TO A SINGULAR INTEGRAL EQUATION

Wenxiong Chen$^1$, Congming Li$^2$, Biao Ou$^3$
$^1$Department of Mathematics
Yeshiva University
New York, NY 10033, USA
e-mail: wchen@ymail.yu.edu
$^2$Department of Applied Mathematics
University of Colorado at Boulder
Boulder, CO 80309, USA
e-mail: cli@colorado.edu
$^3$Department of Mathematics
University of Toledo
Toledo, OH 43606, USA
e-mail: bou@math.utoledo.edu


Abstract.Let $n$ be a positive integer and let $\alpha$ satisfy $ 0 < \alpha < n.$ Consider a positive regular solution $u(x)$ to the integral equation

\begin{displaymath}
u(x) = \int_{R^{n}} \vert x-y\vert^{\alpha -n } u(y)^{\ind }dy.
\end{displaymath}

We use the method of moving planes to prove that for every direction $u(x)$ is symmetric about a plane perpendicular to the direction and monotone on the two sides of the plane. It follows that $u(x)$ is radially symmetric about a point and is a strictly decreasing function of the radius. It then follows that $u(x)$ is a constant multiple of a function of form

\begin{displaymath}
(\frac{t}{t^2 + \vert x-x_{0}\vert^2} )^{(n-\alpha)/2} \,,
\end{displaymath}

where $ t>0, x_{0} \in R^n.$ Our work here adds to and modifies our previous works on the same problem.

Received: March 31, 2009

AMS Subject Classification: 35J99, 45E10, 45G05

Key Words and Phrases: radial symmetry and monotonicity, singular integral, moving planes, inversion transform, Hardy-Littlewood-Sobolev inequalities, regularity

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 52
Issue: 4