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) in R-N, N >= 5, p > N+4/N-4 to be a radially symmetric solution are obtained via the exact asymptotic behavior of u at infinity and the moving plane method (MPM). It is known that above equation admits a unique positive radial entire solution u(x) = u(vertical bar x vertical bar) for any given u(0) > 0, and the asymptotic behavior of u(vertical bar x vertical bar) at infinity is also known. We will see that the behavior similar to that of a radial entire solution of above equation at infinity, 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 infinity is obtained.