This paper is concerned with the problem of prescribing the Paneitz curvature on the standard $n$-sphere, $n\geq 5$. We employ the critical points theory of A. Bahri  to establish an index-counting criteria for existence of solutions when the prescribed function is flat near its critical points for an order $\beta\in (1,n]$.
"The Paneitz curvature problem on $S^n$." Adv. Differential Equations 26 (11/12) 585 - 620, November/December 2021.