  In this article, we tentatively assign the $P_c(4312)$ to be the
$\bar{D}\Sigma_c$ pentaquark molecular state with the spin-parity
$J^P={\frac{1}{2}}^-$, and discuss the factorizable and non-factorizable
contributions in the two-point QCD sum rules for the $\bar{D}\Sigma_c$
molecular state in details to prove the reliability of the single pole
approximation in the hadronic spectral density. We study its two-body strong
decays with the QCD sum rules, special attentions are paid to match the hadron
side with the QCD side of the correlation functions to obtain solid duality. We
obtain the partial decay widths $\Gamma\left(P_c(4312)\to \eta_c
p\right)=0.255\,{\rm{MeV}}$ and $\Gamma\left(P_c(4312)\to J/\psi
p\right)=9.296^{+19.542}_{-9.296}\,\,{\rm{MeV}}$, which are compatible with the
experimental value of the total width, and support assigning the $P_c(4312)$ to
be the $\bar{D}\Sigma_c$ pentaquark molecular state.
