Multiplicity results for constant Q-curvature conformal metrics

We prove that certain subcritical Paneitz-Branson type equations on a closed Riemannian manifold $(M,g)$ have at least $\mathrm{C}\mathrm{a}\mathrm{t}(M)$ positIve solutions, where $\mathrm{C}\mathrm{a}\mathrm{t}(M)$ is the Lusternik-Schnirelmann category of $M$. This implies that if $(X,h)$ is a closed positive Einstein manifold then for $\text{ep}>0$ small enough there are at least $\mathrm{C}\mathrm{a}\mathrm{t}(M)$ metrics of constant $Q$-curvature in the conformal class of the Riemannian product $g+\text{ep}h$.