Existence of Conformal Metrics with Prescribed Q-Curvature

Abstract

We consider the problem of existence of conformal metrics with prescribed Q-curvature on standard sphere $S^n,n\ge 5$. Under the assumption that the order of flatness at critical points of prescribed Q-curvature function $K(x)$ is $\mathrm{\beta \hspace{0.17em}}\in \hspace{0.17em}]1,n-4]$, we give precise estimates on the losses of the compactness, and we prove new existence and multiplicity results through an Euler-Hopf type formula.