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

摘要

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 -Delta u at infinity. We overcome the difficulties due to the fact that e(u) is supercritical for N >= 5, e(u) is not an element of L-N/4 (R-N), and get the right decay rate of u and -Delta u at infinity in order to start the moving plane procedure. (C) 2014 Elsevier Inc. All rights reserved.