Singular Yamabe-type problems with an asymptotically flat metric

Abstract

In this paper, we study the asymptotic symmetry and local behavior of positive solutions at infinity to the equation

outside a bounded set in ${\mathbb{R}}^n$, where $n\ge 3$, $-2<\tau <0$, and $L_g$ is the conformal Laplacian with asymptotically flat Riemannian metric $g$. We prove that the solution, at $\infty$, either converges to a fundamental solution of the Laplace operator on the Euclidean space, or is asymptotically close to a Fowler-type solution defined on ${\mathbb{R}}^n\setminus \{0\}$.