On a perturbed $q$-curvature problem in $S4$

Abstract

Let $g_0$ denote the standard metric on $S^4$ and let $P_{g_0}:={\Delta }_{g_0}^2-2{\Delta }_{g_0}$ denote the corresponding Paneitz operator. In this work, we study the fourth order elliptic problem with exponential nonlinearity

on $S^4$. Here $Q$ is a prescribed smooth function on $S^4$ 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.