# IJPAM: Volume 52, No. 4 (2009)

**POSITIVE REGULAR SOLUTIONS**

TO A SINGULAR INTEGRAL EQUATION

TO A SINGULAR INTEGRAL EQUATION

Wenxiong Chen, Congming Li, Biao Ou

Department of Mathematics

Yeshiva University

New York, NY 10033, USA

e-mail: wchen@ymail.yu.edu

Department of Applied Mathematics

University of Colorado at Boulder

Boulder, CO 80309, USA

e-mail: cli@colorado.edu

Department of Mathematics

University of Toledo

Toledo, OH 43606, USA

e-mail: bou@math.utoledo.edu

Department of Mathematics

Yeshiva University

New York, NY 10033, USA

e-mail: wchen@ymail.yu.edu

Department of Applied Mathematics

University of Colorado at Boulder

Boulder, CO 80309, USA

e-mail: cli@colorado.edu

Department of Mathematics

University of Toledo

Toledo, OH 43606, USA

e-mail: bou@math.utoledo.edu

**Abstract.**Let be a positive integer and let satisfy
Consider a positive regular solution to the integral equation

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

where 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