Equivariant solutions to the optimal partition problem for the prescribed Q-curvature equation

TL;DR

This paper investigates symmetric solutions to the prescribed Q-curvature problem on Einstein manifolds, providing explicit descriptions of solution domains, existence results for solutions, and multiplicity results for sign-changing solutions across related geometric equations.

Contribution

It introduces equivariant methods to analyze the optimal partition problem for Q-curvature equations, extending existence and multiplicity results to new geometric contexts.

Findings

Explicit description of solution domains via group orbits

Existence of least energy symmetric solutions under weaker conditions

Multiplicity of sign-changing solutions for Q-curvature and Yamabe problems

Abstract

We study the optimal partition problem for the prescribed constant $Q$-curvature equation induced by the higher order conformal operators under the effect of cohomogeneity one actions on Einstein manifolds with positive scalar curvature. This allows us to give a precise description of the solution domains and their boundaries in terms of the orbits of the action. We also prove the existence of least energy symmetric solutions to a weakly coupled elliptic system of prescribed $Q$-curvature equations under weaker assumptions and conclude a multiplicity result of sign-changing solutions to the prescribed constant $Q$-curvature problem induced by the Paneitz-Branson operator. Moreover, we study the coercivity of $GJMS$-operators on Ricci solitons, compute the $Q$-curvature of these manifolds, and give a multiplicity result for the sign-changing solutions to the Yamabe problem with prescribed number of nodal domains on the Koiso-Cao Ricci soliton.