  We consider the Paneitz-type equation $\Delta^2 u -\alpha \Delta u +\beta
(u-u^q ) =0$ on a closed Riemannian manifold $(M,g)$. We reduce the equation to
a fourth-order ordinary differential equation assuming that $(M,g)$ admits a
proper isoparametric function. Assuming that $\alpha$ and $\beta$ are positive
and $\alpha^2 >4\beta$, we prove that the global nonconstant solutions of this
ordinary differential equation only has nondegenerate critical points. Applying
global bifurcation theory we prove multiplicity results for positive solutions
of the equation. As an application and motivation we prove multiplicity results
for conformal constant $Q$-curvature metrics. For example, consider closed
positive Einstein manifolds $(M^n ,g )$ and $(X^m , h)$ of dimensions $n,m \geq
3$. Assuming that $M$ admits a proper isoparametric function (with a symmetry
condition) we prove that as $\delta >0$ gets closer to 0, the number of
constant $Q$-curvature metrics conformal to $g_{\delta} = g+\delta h$ goes to
infinity.
