Radial symmetry of entire solutions of a bi-harmonic equation with exponential nonlinearity

We obtain necessary and sufficient conditions for an entire solution u of a bi-harmonic equation with exponential nonlinearity e^u to be a radially symmetric solution. The standard tool to obtain the radial symmetry for a system of equations is the moving plane method (MPM). In order to apply the MPM, we need to know the asymptotic expansions of u and -δu at ∞. We overcome the difficulties due to the fact that e^u is supercritical for N≥5, eu∉LN4(RN), and get the right decay rate of u and -δu at ∞ in order to start the moving plane procedure.