On the perturbed Q -curvature problem on S4S4

Let g0g0 denote the standard metric on S4S4 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 S4S4 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.