On the prescribed Q-curvature problem on [formula omitted

Abstract

In this paper we prescribe a fourth order curvature – the Q-curvature on the standard n-sphere, n ⩾ 5 . Under flatness condition of order β, n − 4 ⩽ β < n near each critical point of the prescribed Q-curvature function, we characterize the critical points at infinity of the associated variational problem and we prove new existence results through Euler–Hopf formulae type. Our argument gives an upper bound on the Morse index of the obtained solutions. We also give a lower bound on the number of conformal metrics having the same Q-curvature.