The Paneitz curvature problem on $S^n$

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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 [3] to establish an index-counting criteria for existence of solutions when the prescribed function is flat near its critical points for an order $\beta\in (1,n]$.