  We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem
in domains with conic singularities and establish asymptotic expansions with
respect to two normal directions simultaneously. 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. Key step is the study of the eigenvalues growth and
eigenfunctions estimates.
