Based on the effective Lagrangian in the heavy quark limit, we calculate the one-boson-exchange interaction kernel of $P_{\psi }^N{\left(4312\right)}^{+}$ $$ {P}_{\psi}^N{(4312)}^{+} $$ as the $\overline{D}{\Sigma }_c$ $$ \overline{D}{\Sigma}_c $$ molecular state in isospin- $\frac{1}{2}$ $$ \frac{1}{2} $$ . We present the Bethe-Salpeter equation and wave function for the constituent particles to be a (pseudo)scalar meson and a $\frac{1}{2}$ $$ \frac{1}{2} $$ baryon. By solving the Bethe-Salpeter equation, we obtain $P_{\psi }^N{\left(4312\right)}^{+}$ $$ {P}_{\psi}^N{(4312)}^{+} $$ as the $\overline{D}{\Sigma }_c$ $$ \overline{D}{\Sigma}_c $$ molecular state with J P = ${\left(\frac{1}{2}\right)}^{-}$ $$ {\left(\frac{1}{2}\right)}^{-} $$ . Combining the effective Lagrangian and the obtained BS wave function, the partial decay widths of $P_{\psi }^N{\left(4312\right)}^{+}$ $$ {P}_{\psi}^N{(4312)}^{+} $$ to J/ψp, η c p, ${\overline{D}}^{\ast 0}{\Lambda }_c^{+}$ $$ {\overline{D}}^{\ast 0}{\Lambda}_c^{+} $$ and ${\overline{D}}^0{\Lambda }_c^{+}$ $$ {\overline{D}}^0{\Lambda}_c^{+} $$ are calculated to be 0.17, 0.085, 8.8, and 0.026 MeV, respectively, which are roughly consistent with the LHCb experimental measurements and some other theoretical researches. The obtain results indicate the fraction of ${\overline{D}}^{\ast 0}{\Lambda }_c^{+}$ $$ {\overline{D}}^{\ast 0}{\Lambda}_c^{+} $$ channel amounts to ~ 90% of $P_{\psi }^N{\left(4312\right)}^{+}$ $$ {P}_{\psi}^N{(4312)}^{+} $$ , and is a highly promising channel to be discovered in the near future experiments. Our results favor the interpretation of $P_{\psi }^N{\left(4312\right)}^{+}$ $$ {P}_{\psi}^N{(4312)}^{+} $$ as the $\overline{D}{\Sigma }_c$ $$ \overline{D}{\Sigma}_c $$ molecular state with J P = ${\left(\frac{1}{2}\right)}^{-}$ $$ {\left(\frac{1}{2}\right)}^{-} $$ and isospin I = $\frac{1}{2}$ $$ \frac{1}{2} $$ .