  We study the possible bound states of the $D_1D$ system in the Bethe-Salpeter
(BS) formalism in the ladder and instantaneous approximations. By solving the
BS equation numerically with the kernel containing one-particle exchange
diagrams and introducing three different form factors (monopole, dipole, and
exponential form factors) at the vertices, we investigate whether the isoscalar
and isovector $D_1D$ bound states may exist, respectively. We find that
$Y(4260)$ could be accommodated as a $D_1D$ molecule, whereas the
interpretation of $Z_2^+(4250)$ as a $D_1D$ molecule is disfavored. The bottom
analog of $Y(4260)$ may exist but that of $Z_2^+(4250)$ does not.
