Toward Best Isoperimetric Constants for $(H^1,BMO)$-Normal Conformal Metrics on $mathbb R^n$, $nge 3$

The aim of this article is: (a) To establish the existence of the best isoperimetric constants for the $(H^1,BMO)$-normal conformal metrics $e^{2u}|dx|^2$ on $mathbb R^n$, $nge 3$, i.e., the conformal metrics with the Q-curvature orientated conditions $$ (-Delta)^{n/2}uin H^1(mathbb R^n) & u(x)=hbox{const.}+frac{int_{mathbb R^n}(logfrac{|cdot|}{|x-cdot|})(-Delta)^{n/2} u(cdot) dmathcal{H}^n(cdot)}{2^{n-1}pi^{n/2}Gamma(n/2)}; $$ (b) To prove that $(nomega_n^frac1n)^frac{n}{n-1}$ is the optimal upper bound of the best isoperimetric constants for the complete $(H^1,BMO)$-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 Greens functions for the $n$-Laplacian operators $-hbox{div}(| abla u|^{n-2} abla u)$.