Classification of solutions of certain fourth-order nonlinear elliptic equations in R{double-struck}4

Abstract

We consider the uniqueness of solutions of the equation Δ2 u = eu in fourdimensional Euclidean space. Our main result is that the solutions are all classical ones, provided that the energy of the solutions is finite and the diffusion of the solutions decays to zero at infinity. The method we used in this paper is known as the method of moving spheres.