  We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem
in finite cones and establish optimal asymptotic expansions in terms of the
corresponding solutions in infinite cones. The spherical domains over which
cones are formed are allowed to have singularities. An elliptic operator on
such spherical domains with coefficients singular on boundary play an important
role. Due to the singularity of the spherical domains, extra cares are needed
for the study of the global regularity of the eigenfunctions and solutions of
the associated singular Dirichlet problem.
