On isolated singularities of the conformal Gaussian curvature equation and $Q$-curvature equation

Abstract

In this paper, we study the isolated singularities of the conformal Gaussian curvature equation \[ -Δu = K(x) e^{u} \quad ~ in ~ B_{1} \setminus \{ 0 \}, \] where $B_1\setminus \{0\}\subset {\mathbb{R}}^2$ is the punctured unit disc. Under the assumption that the Gaussian curvature $K\in L^{\infty }(B_1)$ is nonnegative, we establish the asymptotic behavior of solutions near the singularity. When $K\equiv 1$, a similar result has been obtained by Chou and Wan (Pacific J. Math. 1994) using the method of complex analysis. Our proof is entirely based on the PDE method and applies to the general Gaussian curvature $K(x)$. Furthermore, our approach is also available for characterizing isolated singularities of the conformal $Q$-curvature equation $(-\Delta {)}^{\frac{n}{2}}u=K(x)e^u$ in any dimension $n\ge 3$. This equation arises from the prescribing $Q$-curvature problem.

20 pages