Radial Symmetry of Entire Solutions of a Biharmonic Equation with Supercritical Exponent

Necessary and sufficient conditions for a regular positive entire solution u of a biharmonic equation\Delta^{2}u=u^{p}\quad\text{in }\mathbb{R}^{N},\,N\geq 5,\,p>\frac{N+4}{N-4}to be a radially symmetric solution are obtained via the exact asymptotic behavior of u at {\infty} and the moving plane method (MPM). It is known that above equation admits a unique positive radial entire solution {u(x)=u(|x|)} for any given {u(0)>0}, and the asymptotic behavior of {u(|x|)} at {\infty} is also known. We will see that the behavior similar to that of a radial entire solution of above equation at {\infty}, in turn, determines the radial symmetry of a general positive entire solution {u(x)} of the equation. To make the procedure of the MPM work, the precise asymptotic behavior of u at {\infty} is obtained.

Entire Solutions of an Integral Equation in R5

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.