Global bifurcation for Paneitz type equations and constant Q-curvature metrics (Q6616060)

The \(Q\)-curvature is a conformal invariant that can be viewed as an analogue of the Gaussian curvature in dimension 2, due to its connection with the topology of the underlying manifold via the Gauss-Bonnet-Chern formula in dimension 4. In higher dimensions, \(Q\)-curvature is defined using the Paneitz operator, a conformally covariant differential operator of order \(4\). In dimension \(N\geq 5\) it is defined by:\N\[\NP_{\mathfrak{g}}=\Delta^2 u-\operatorname{div}(a_N S\cdot g+b_N\mathrm{Ric}_{\mathfrak{g}})du+\frac{N-4}{2}Q_{\mathfrak{g}},\N\]\Nfor some dimensional constants \(a_N\) and \(b_N\). The \(Q\)-curvature is then the scalar part of \(P_{\mathfrak{g}}\), that is, \(\displaystyle{Q_{\mathfrak{g}}=\frac{1}{N-4}}.\)\N\NThe \(Q\)-curvature problem asks for the existence of conformal metrics with prescribed constant \(Q\)-curvature, which is equivalent to the existence of positive solutions to the equation:\N\[\NP_{\mathfrak{g}_o}=\lambda u^{\frac{N+4}{N-4}}, \ \lambda\in\mathbb{R}.\N\]\N\NIn this paper, the authors study a Paneitz-type equation of the form:\N\[\N\Delta^2u-\alpha\Delta u+\beta(u-u^q)=0\N\]\Non manifolds Riemannian manifolds of dimension \(N\geq 5\).\N\NThe existence of a proper isoparametric function \(F\) is assumed. By restricting to functions that are constant along the level sets of \(F\), the previous equation reduces to an ODE, allowing for a one-dimensional reduction. An isoparametric function has only two critical points, and its critical levels are minimal submanifolds, called focal manifolds. Then it is needed to establish an appropriate asymptotic extension of functions to the focal manifolds.\N\NThe authors investigate the bifurcation of the family of trivial solutions arising from the one-dimensional reduction of the aforementioned Paneitz-type equation. Theorem 1.1 is essential in the application of bifurcation theory and is used to establish that the number of critical points remains constant within each connected component. Theorem 1.2 obtains the multiplicity of solutions to the reduced equation, with arbitrary critical points. In Theorem 1.3, it is proved that there are no positive nonconstant solutions for \(\alpha, \beta\) close to zero, and \(q<p^*\). Here \(p^*=\frac{N+4}{N-4}\) if \(N\geq 5\); and \(p^*=\infty\) if \(N=3, 4\).\N\NAs an application, the authors obtain multiplicity results for constant \(Q\)-curvature metrics on products of Einstein manifolds \(M\times X\) with positive Ricci curvature, and where \(M\) admits a cohomogeneity one action. This is done in Corollaries 1.4 and 1.5. Corollary 1.6 concerns an explicit case where \(M\) is the 3-sphere with the round metric, using the actions by \(O(3)\) and \(O(2)\times O(2)\).