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

TLDR: In this article, a necessary and sufficient condition for solutions obtained from the smooth conformal metrics on S 4>>\s was established for the stereograph projection of the biharmonic operator.

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 $ \qquad for $ n \ge5 $, \cr} $ (2) where $ \Delta^2 $ denotes the biharmonic operator in R n . 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 .