Global bifurcation for Paneitz type equations and constant Q-curvature metrics
type: Item
instance of: scholarly article
title: Global bifurcation for Paneitz type equations and constant Q-curvature metrics
author: Jurgen Julio-Batalla
author: Jimmy Petean
published in: Journal of Differential Equations
publication date: 2024-10-08T00:00:00Z
review text: 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. …
reviewed by: Jonatán Torres Orozco
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58J55
Mathematics Subject Classification ID: 35J30
zbMATH DE Number: 7923707
zbMATH Keywords: Paneitz equation
zbMATH Keywords: Q-curvature
zbMATH Keywords: bifurcation theory
MaRDI profile type: Publication
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DOI: 10.1016/J.JDE.2024.07.026
description: scientific article; zbMATH DE number 7923707