Compactness of conformal metrics with constant $Q$-curvature. I

摘要: We study compactness for nonnegative solutions of the fourth order constant $Q$-curvature equations on smooth compact Riemannian manifolds dimension $\ge 5$. If equals $-1$, we prove that all are universally bounded. is $1$, assuming Paneitz operator's kernel trivial and its Green function positive, establish universal energy bounds which either locally conformally flat (LCF) or $\le 9$. Moreover, in addition a positive mass type theorem holds operator, $C^4$. Positive theorems have been verified recently LCF 7$, when Yamabe invariant positive. also that, 8$, Weyl tensor has to vanish at possible blow up points sequence blowing solutions. This implies result 8$ does not anywhere. To overcome difficulties stemming from elliptic equations, develop analysis procedure via integral equations.