Table of Contents
ISRN Mathematical Analysis
Volume 2013, Article ID 384394, 17 pages
http://dx.doi.org/10.1155/2013/384394
Research Article

Entire Solutions of an Integral Equation in

Department of Mathematics, National University of Singapore, Block S17 (SOC1), 10 Lower Kent Ridge Road, Singapore 119076

Received 29 May 2013; Accepted 17 June 2013

Academic Editors: G. Gripenberg, M. McKibben, and M. Winter

Copyright © 2013 Xin Feng and Xingwang Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We will study the entire positive solution of the geometrically and analytically interesting integral equation: with in . We will show that only when , there are positive entire solutions which are given by the closed form up to dilation and translation. The paper consists of two parts. The first part is devoted to showing that must be equal to 11 if there exists a positive entire solution to the integral equation. The tool to reach this conclusion is the well-known Pohozev identity. The amazing cancelation occurred in Pohozev’s identity helps us to conclude the claim. It is this exponent which makes the moving sphere method work. In the second part, as normal, we adopt the moving sphere method based on the integral form to solve the integral equation.