  It is natural to consider continuous dependence of the $n$-th eigenvalue on
$d$-dimensional ($d\geq2$) Sturm-Liouville problems after the results on
$1$-dimensional case by Kong, Wu and Zettl [14]. In this paper, we find all the
boundary conditions such that the $n$-th eigenvalue is not continuous, and give
complete characterization of asymptotic behavior of the $n$-th eigenvalue. This
renders a precise description of the jump phenomena of the $n$-th eigenvalue
near such a boundary condition. Furthermore, we divide the space of boundary
conditions into $2d+1$ layers and show that the $n$-th eigenvalue is
continuously dependent on Sturm-Liouville equations and on boundary conditions
when restricted into each layer. In addition, we prove that the analytic and
geometric multiplicities of an eigenvalue are equal. Finally, we obtain
derivative formula and positive direction of eigenvalues with respect to
boundary conditions.
