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} Δ^2 u = 6 e⁴ᵘ & in R^4, e⁴ᵘ ∈ L^1(R^4), {cases} (1) and {cases} Δ^2 u = un+4 n-4, u>0 in { R}^n for n ≥5 , {cases} (2) where Δ^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 .