  We constructed an effective tight-binding model with five Cr $3d$ orbitals
for LaOCrAs according to first-principles calculations. Basing on this model,
we investigated possible superconductivity induced by correlations in doped
LaOCrAs using the functional renormalization group (FRG). We find that there
are two domes of superconductivity in electron-doped LaOCrAs. With increasing
electron doping, the ground state of the system evolves from G-type
antiferromagnetism in the parent compound to an incipient $s_\pm$-wave
superconducting phase dominated by electron bands derived from the
$d_{3z^2-r^2}$ orbital as the filling is above $4.2$ electrons per site on the
$d$-orbitals of Cr. The gap function has strong octet anisotropy on the Fermi
pocket around the zone center and diminishes on the other pockets. In electron
over-doped LaOCrAs, the system develops $d_{x^2-y^2}$-wave superconducting
phase and the active band derives from the $d_{xy}$ orbital. Inbetween the two
superconducting domes, a time-reversal symmetry breaking $s+id$ SC phase is
likely to occur. We also find $s_\pm$-wave superconducting phase in the
hole-doped case.
