Local estimates for conformal $Q$-curvature equations

Abstract

We derive local estimates of positive solutions to the conformal $Q$-curvature equation $$ (-Δ)^m u = K(x) u^{\frac{n+2m}{n-2m}} ~~~~~~ in ~ Ω\backslash Λ$$ near their singular set $Λ$, where $Ω\subset \mathbb{R}^n$ is an open set, $K(x)$ is a positive continuous function on $Ω$, $Λ$ is a closed subset of $\mathbb{R}^n$, $2 \leq m < n/2$ and $m$ is an integer. Under certain flatness conditions at critical points of $K$ on $Λ$, we prove that $u(x) \leq C [{dist}(x, Λ)]^{-(n-2m)/2}$ when the upper Minkowski dimension of $Λ$ is less than $(n-2m)/2$.