A classification of solutions of a conformally invariant fourth order equation in Rn

Abstract

In this paper, we consider the following conformally invariant equations of fourth order¶ \( \cases {\Delta^2 u = 6 e^{4u} &in \)\bf {R}^4,\( \cr e^{4u} \in L^1(\bf {R}^4),\cr} \)(1)¶and¶ \( \cases {\Delta^2 u = u^{n+4 \over n-4}, \cr u>0 & in \) {\bf R}^n $for$ n \ge5 \( , \cr} \) (2) where ${\Delta }^2$ denotes the biharmonic operator in Rn. By employing the method of moving planes, we are able to prove that all positive solutions of (2) are arised from the smooth conformal metrics on S n by the stereograph projection. For equation (1), we prove a necessary and sufficient condition for solutions obtained from the smooth conformal metrics on S 4.