  This paper deals with the conformal deformation of the standard metric in a
domain on the sphere to a complete metric with the constant scalar curvature.
The problem of description of domains allowing such deformation originates in
the works of Loewner and Nirenberg, and Schoen and Yau concerned with the
locally conformally flat manifolds. The goal of this work is to apply ideas
from the nonlinear potential theory to the problem. They allow, in particular,
to solve the problem in the case of the constant negative scalar curvature.
