Abstract
We will study the entire positive C0 solution of the geometrically and analytically interesting integral equation: u(x)=1/C5∫R5|x-y|u-q(y)dy with 0<q in R5. We will show that only when q=11, there are positive entire solutions which are given by the closed form u(x)=c(1+|x|2)1/2 up to dilation and translation. The paper consists of two parts. The first part is devoted to showing that q 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.
Highlights
In this paper, we will study a very special type of the integral equation
The moving plane method can be applied to prove the radial symmetry of solutions, and one only needs to classify radial solutions
The method of moving planes in the integral form has been developed by Chen and Li [4]
Summary
We will study the entire positive C0 solution of the geometrically and analytically interesting integral equation: u(x) = 1/C5 ∫R5 |x − y|u−q(y)dy with 0 < q in R5. We will show that only when q = 11, there are positive entire solutions which are given by the closed form u(x) = c(1 + |x|2)1/2 up to dilation and translation. The first part is devoted to showing that q must be equal to 11 if there exists a positive entire solution to the integral equation. 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. As normal, we adopt the moving sphere method based on the integral form to solve the integral equation
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
