On the prescribed Q-curvature problem on [formula omitted

Abstract

In this Note we prescribe a fourth order curvature – the Q-curvature on the standard n-sphere, n ⩾ 5 . Under the “flatness condition” of order β, n − 4 ⩽ β < n near each critical point of the prescribed Q-curvature function, we prove new existence result through an Euler–Hopf type formula. Our argument gives a lower bound on the number of conformal metrics having the same Q-curvature.