Global bifurcation for Paneitz type equations and constant Q-curvature metrics

Abstract

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 $n,m\ge 3$. 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.