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

In this paper, we consider the following conformally invariant equations of fourth order cases ² u = 6 e^4u & in R⁴, e^4u L¹ (R⁴), cases (1) and cases ² u = u^n+4 n-4, u>0 in Rⁿ for n 5, cases (2) where 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 Sⁿ by the stereograph projection. For equation (1), we prove a necessary and sufficient condition for solutions obtained from the smooth conformal metrics on S⁴.