  The recent discovery of possible high temperature superconductivity in single
crystals of $La_3Ni_2O_7$ under pressure renews the interest in research on
nickelates. The DFT calculations reveal that both $d_{z^2}$ and $d_{x^2-y^2}$
orbitals are active, which suggests a minimal two-orbital model to capture the
low-energy physics of this system. In this work, we study a bilayer two-orbital
$t-J$ model within multiband Gutzwiller approximation, and discuss the
magnetism as well as the superconductivity over a wide range of the hole
doping. Owing to the inter-orbital super-exchange process between $d_{z^2}$ and
$d_{x^2-y^2}$ orbitals, the induced ferromagnetic coupling within layers
competes with the conventional antiferromagnetic coupling, and leads to
complicated hole doping dependence for the magnetic properties in the system.
With increasing hole doping, the system transfers to A-AFM state from the
starting G-AFM state. We also find the inter-layer superconducting pairing of
$d_{x^2-y^2}$ orbitals dominates due to the large hopping parameter of
$d_{z^2}$ along the vertical inter-layer bonds and significant Hund's coupling
between $d_{z^2}$ and $d_{x^2-y^2}$ orbitals. Meanwhile, the G-AFM state and
superconductivity state can coexist in the low hole doping regime. To take
account of the pressure, we also analyze the impacts of inter-layer hopping
amplitude on the system properties.
