Toward best isoperimetric constants for (H1,BMO)(H1,BMO)-normal conformal metrics on RnRn, n⩾3n⩾3

The aim of this article is: (a) to establish the existence of the best isoperimetric constants for the (H1,BMO)(H1,BMO)-normal conformal metrics e2u|dx|2e2u|dx|2 on RnRn, n⩾3n⩾3, i.e., the conformal metrics with the Q-curvature orientated conditions(−Δ)n/2u∈H1(Rn)andu(x)=const.+∫Rn(log|⋅||x−⋅|)(−Δ)n/2u(⋅)dHn(⋅)2n−1πn/2Γ(n/2); (b) to prove that (nωn1n)nn−1 is the optimal upper bound of the best isoperimetric constants for the complete (H1,BMO)(H1,BMO)-normal conformal metrics with nonnegative scalar curvature; (c) to find the optimal upper bound of the best isoperimetric constants via the quotients of two power integrals of Green's functions for the n  -Laplacian operators −div(|∇u|n−2∇u)−div(|∇u|n−2∇u).