Perturbations of Paneitz-Branson operators on S^n

We prove the existence of solutions on the standard unit sphere $(S^n,h)$ for the equation $P_h^n u=d_n\mid u \mid^{\frac{8}{n-4}}u+(\epsilon K+o(\epsilon))\mid u \mid ^{q-1}u$, $\epsilon$ small, and $1\leq q \leq \frac{n+4}{n-4}$, where $P_g^n$ is the fourth order conformally invariant Paneitz-Branson operator. We will approach this problem via a finite dimensional reduction which lead us to consider the "stable" critical points of the "Melnikov function": in the case $q=\frac{n+4}{n-4}$ a more subtle analysis will be carried out by means of a Morse relation for functions on manifolds with boundary which are quite degenerate on the boundary.