Local estimates for conformal $Q$-curvature equations

We derive local estimates of positive solutions to the conformal $Q$-curvature equation $$ (-Delta)^m u = K(x) u^{frac{n+2m}{n-2m}} ~~~~~~ in ~ Omega backslash Lambda $$ near their singular set $Lambda$, where $Omega subset mathbb{R}^n$ is an open set, $K(x)$ is a positive continuous function on $Omega$, $Lambda$ 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 $Lambda$, we prove that $u(x) leq C [{dist}(x, Lambda)]^{-(n-2m)/2}$ when the upper Minkowski dimension of $Lambda$ is less than $(n-2m)/2$.