Existence of conformal metrics on $S^n$ with prescribed fourth-order invariant

Abstract

In this paper we prescribe a fourth-order conformal invariant on the standard $n$-sphere, with $ n\geq5 $, and study the related fourth-order elliptic equation. We first find some existence results in the perturbative case. After some blow-up analysis we build a homotopy to pass from the perturbative case to the nonperturbative one under some flatness condition. Finally we state some existence results under the assumption of symmetry.