Local behavior of positive solutions of higher order conformally invariant equations with a singular set

Abstract

We study some properties of positive solutions to the higher order conformally invariant equation with a singular set

$$\begin{aligned} (-\Delta )^m u = u^{\frac{n+2m}{n-2m}} ~~~~~~ \text {in} ~ \Omega \backslash \Lambda , \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^n$ is an open domain, $\Lambda$ is a closed subset of ${\mathbb {R}}^n$, $1 \le m < n/2$ and m is an integer. We first establish a local estimate for a singular positive solution u near its singular set $\Lambda$ when $\Lambda \subset \Omega$ is a compact set with the upper Minkowski dimension $\overline{\text {dim}}_M(\Lambda ) < \frac{n-2m}{2}$, or $\Lambda \subset \Omega$ is a smooth k-dimensional closed manifold with $k\le \frac{n-2m}{2}$. Furthermore, we also show the asymptotic symmetry of singular positive solutions supposing $\Lambda \subset \Omega$ is a smooth k-dimensional closed manifold with $k\le \frac{n-2m}{2}$. Finally, a global symmetry result for solutions is obtained when $\Omega$ is the whole space and $\Lambda$ is a k-dimensional hyperplane with $k\le \frac{n-2m}{2}$.