Cooperation between electron-phonon coupling and electronic interaction in bilayer nickelates La$_3$Ni$_2$O$_7$

The recent observation of high-T$_c$ superconductivity in the bilayer nickelate La$_3$Ni$_2$O$_7$ under pressure has garnered significant interests. While researches have predominantly focused on the role of electron-electron interactions in the superconducting mechanism, the impact of electron-phonon coupling (EPC) has remained elusive. In this work, we perform first-principles calculations to study the phonon spectrum and electron-phonon coupling within La$_3$Ni$_2$O$_7$ under pressure and explore of the interplay between EPC and electronic interactions on the superconductivity by employing functional renormalization group approach. Our calculations reveal that EPC alone is insufficient to trigger superconductivity in La$_3$Ni$_2$O$_7$ under pressure. We identify unique out-of-plane and in-plane breathing phonon modes which selectively couple with the Ni $d_{z^2}$ and $d_{x^2-y^2}$ orbitals, showcasing an orbital-selective EPC. Within the bilayer two-orbital model, it is revealed that solely electronic interactions foster $s_{\pm}$-wave pairing characterized by notable frustration in the band space, leading to a low transition temperature. Remarkably, we find that this out-of-plane EPC can act in concert with electronic interactions to promote the onsite and interlayer pairing in the $d_{z^2}$ orbital, partially releasing the pairing frustration and thus elevating T$_c$. In contrast, the inclusion of in-plane EPC only marginally affects the superconductivity, distinct from the cuprates. Potential experimental implications in La$_3$Ni$_2$O$_7$ are also discussed.