  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 infinite
projected 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 $d_{x^2-y^2}$ electrons
exhibit 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 $J_\perp$, while
the inherent intralayer $t_\parallel$ and $J_\parallel$ are small. Based on the
iPEPS 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 by
the interlayer coupling $J_\perp$; while the $d_{z^2}$ band shows relatively
weak SC order as a result of small $t_\parallel$ (lack of coherence) but large
$t_\perp$ (strong Pauli blocking). Furthermore, by substituting rare-earth
element Pm or Sm with La, we find an enhanced SC order, which opens up a
promising avenue for discovering nickelate superconductors with even higher
$T_c$.
