  In the present work, the partial decay widths of $P_{c}(4312)$ to $\eta_{c}
p$ and $J/\psi p$ are investigated with the QCD sum rule method under the
assumption that $P_{c}(4312)$ is a $\bar{D}\Sigma_{c}$ molecular state with
$J^{P}=\frac{1}{2}^{-}$. In the analysis, the pole residue of $P_{c}(4312)$,
one of the input parameters for the calculations of the strong decay constants,
is calculated first. With the numerical values of the strong decay constants,
the partial decay widths to $\eta_{c} p$ and $J/\psi p$ are estimated to be
$\Gamma(P_{c}(4312)\rightarrow \eta_{c} p)=5.54^{+0.75}_{-0.5}\mbox{MeV}$ and
$\Gamma(P_{c}(4312)\rightarrow J/\psi p)=1.67^{+0.92}_{-0.56}\mbox{MeV}$,
respectively, which are compatible with the measured total width of
$P_{c}(4312)$. The results suggest that it is reasonable to assign
$P_{c}(4312)$ to be a $\bar{D}\Sigma_{c}$ molecular state with
$J^{P}=\frac{1}{2}^{-}$.
