A classification of solutions of a conformally invariant fourth order equation in $\mathbf{R}^n$

Abstract

In this paper, we consider the following conformally invariant equations of fourth order \begin{cases} \Delta^2 u = 6 e^{4u} & \text{in } \mathbf {R}^4, \cr e^{4u} \in L^1(\mathbf {R}^4), \end{cases} \qquad (1) and \begin{cases} \Delta^2 u = u^{n+4 \over n-4}, \cr u>0 \quad \text{in } {\mathbf R}^n \quad \text{for } n \ge5 , \cr \end{cases} \quad (2) where \Delta^2 denotes the biharmonic operator in \mathbf{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 .