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The newly discovered high-$T_c$ nickelate superconductor La$_3$Ni$_2$O$_7$has generated significant research interest. To uncover the pairing mechanism,it is essential to investigate the intriguing interplay between the two $e_g$,i.e., $d_{x^2-y^2}$ and $d_{z^2}$ orbitals. Here we perform an infiniteprojected entangled-pair state (iPEPS) study of the bilayer $t$-$J$ model,directly in the thermodynamic limit and with orbitally selective parameters for$d_{x^2-y^2}$ and $d_{z^2}$ orbitals, respectively. The $d_{x^2-y^2}$ electronsexhibit significant intralayer hopping $t_\parallel$ and spin couplings$J_\parallel$, with interlayer $J_\perp$ passed from the $d_{z^2}$ electrons.However, the interlayer $t_\perp$ is negligible in this case. In contrast, the$d_{z^2}$ orbital demonstrates strong interlayer $t_\perp$ and $J_\perp$, whilethe inherent intralayer $t_\parallel$ and $J_\parallel$ are small. Based on theiPEPS results, we find clear orbital-selective behaviors in La$_3$Ni$_2$O$_7$.The $d_{x^2-y^2}$ orbitals exhibit robust superconductive (SC) order driven bythe interlayer coupling $J_\perp$; while the $d_{z^2}$ band shows relativelyweak SC order even if we artificially introduce some $t_\parallel$ to it.Furthermore, by substituting rare-earth element Pm or Sm with La, we find anenhanced SC order, which opens up a promising avenue for discovering nickelatesuperconductors with even higher $T_c$.

Orbital-selective Superconductivity in the Pressurized Bilayer Nickelate La$_3$Ni$_2$O$_7$: An Infinite Projected Entangled-Pair State Study