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We study the optimal partition problem for the prescribed constant$Q$-curvature equation induced by the higher order conformal operators underthe effect of cohomogeneity one actions on Einstein manifolds with positivescalar curvature. This allows us to give a precise description of the solutiondomains and their boundaries in terms of the orbits of the action. We alsoprove the existence of least energy symmetric solutions to a weakly coupledelliptic system of prescribed $Q$-curvature equations under weaker assumptionsand conclude a multiplicity result of sign-changing solutions to the prescribedconstant $Q$-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 resultfor the sign-changing solutions to the Yamabe problem with prescribed number ofnodal domains on the Koiso-Cao Ricci soliton.

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