Local estimates for conformal $Q$-curvature equations

TL;DR

This paper establishes local estimates for positive solutions to conformal $Q$-curvature equations near singular sets, under flatness and dimension conditions, extending understanding of solution behavior in geometric analysis.

Contribution

It provides new local estimates for solutions near singularities in conformal $Q$-curvature equations, considering the flatness of the function $K$ and the dimension of the singular set.

Findings

Solutions are bounded by a power of the distance to the singular set.

Estimates depend on the flatness of $K$ at critical points.

Results hold when the Minkowski dimension of the singular set is below a threshold.

Abstract

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 $(n-2m)/2$.