Asymptotic analysis for fourth order Paneitz equations with critical growth

TL;DR

This paper analyzes the asymptotic behavior of solutions to fourth order Paneitz equations with critical growth on conformally flat manifolds, revealing stability, compactness, and criticality properties in geometric analysis.

Contribution

It provides sharp asymptotic estimates and establishes stability and compactness results for solutions of Paneitz equations in higher dimensions, highlighting their critical nature.

Findings

Derived sharp asymptotics for solutions

Established stability and compactness properties

Identified the criticality of the geometric equation

Abstract

We investigate fourth order Paneitz equations of critical growth in the case of $n$-dimensional closed conformally flat manifolds, $n \ge 5$. Such equations arise from conformal geometry and are modelized on the Einstein case of the geometric equation describing the effects of conformal changes of metrics on the $Q$-curvature. We obtain sharp asymptotics for arbitrary bounded energy sequences of solutions of our equations from which we derive stability and compactness properties. In doing so we establish the criticality of the geometric equation with respect to the trace of its second order terms.