Multiplicity results for constant Q-curvature conformal metrics

Abstract

In this paper we provide a positive lower bound for the number of metrics of constant Q-curvature which are conformal to a Riemannian product of the form \((M\times X, g+\delta h)\), where \(\delta >0\) is a small positive constant, (M, g) is a closed (compact without boundary) n-dimensional Riemannian manifold and (X, h) a closed m-dimensional (positive) Einstein manifold. We assume that \(m\ge 3\) and \(n\ge 2\) or, if \(m=2\), that \(n\ge 7\). More specifically, we study the constant Q-curvature equation on the Riemannian product \((M\times X, g+\delta h)\), which becomes, by restricting the equation to functions which depend only on the M-variable, a subcritical equation on (M, g) driven by a fourth order operator, known as the Paneitz operator. Then we prove that, for \(\delta >0\) small enough, the equation has at least \(\textrm{Cat}(M)\) positive solutions, where \(\textrm{Cat}(M)\) is the Lusternik-Schnirelmann category of M. This implies that there are at least \(\textrm{Cat}(M)\) metrics of constant Q-curvature in the conformal class of the Riemannian product \((M\times X, g+\delta h)\).