Let g0 denote the standard metric on S4 and Pg0=δg02-2δg0 denote the corresponding Paneitz operator. In this work, we study the following fourth order elliptic problem with exponential nonlinearityPg0u+6=2Q(x)e4uon S4. Here Q is a prescribed smooth function on S4 which is assumed to be a perturbation of a constant. We prove existence results to the above problem under assumptions only on the "shape" of Q near its critical points. These are more general than the non-degeneracy conditions assumed so far. We also show local uniqueness and exact multiplicity results for this problem. The main tool used is the Lyapunov-Schmidt reduction. © 2013.