Classification of solutions of certain fourth-order nonlinear elliptic equations in $\mathbb R^4$

Xingwang Xu

Vol. 225 (2006), No. 2, 361–378

DOI: 10.2140/pjm.2006.225.361

Abstract

We consider the uniqueness of solutions of the equation ${\Delta }^{2}u=e^u$ in four-dimensional 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.

Keywords

conformally invariant equation, symmetry, moving sphere method

Mathematical Subject Classification 2000

Primary: 35J60

Secondary: 53C21

Milestones

Received: 19 August 2004

Revised: 14 October 2004

Accepted: 14 October 2004

Published: 1 June 2006