  In this work, we investigate possible bound states of the
$\Lambda_c\bar{\Lambda}_c$ system in the Bethe-Salpeter formalism in the ladder
and instantaneous approximations. By numerically solving the Bethe-Salpeter
equation, we confirm the existence of $\Lambda_c\bar{\Lambda}_c$ bound states
with quantum numbers $J^{PC}=0^{-+}$ and $J^{PC}=1^{--}$. We further
investigate the partial decay widths of the $\Lambda_c\bar{\Lambda}_c$ bound
states into $N\bar{N}$, $D\bar{D}$, $D\bar{D}^\ast$, $D^\ast\bar{D}^\ast$,
$\pi\bar{\pi}$, and $K\bar{K}$. Our results indicate that the decay width of
the $\Lambda_c\bar{\Lambda}_c$ bound state with $J^{PC}=1^{--}$ is much larger
than that with $J^{PC}=0^{-+}$, and among their decay channels, the
$D\bar{D}^\ast$ final state is the main decay mode. We suggest experiments to
search for the $\Lambda_c\bar{\Lambda}_c$ bound states in the $D\bar{D}^\ast$
final state.
