  Multi-orbital quantum materials with strong interactions can host a variety
of novel phases. In this work we study the possibility of interaction-driven
superconductivity in the iridate compound Sr$_2$IrO$_4$ under strain and
doping. We find numerous regimes of strain-induced superconductivity in which
the pairing structure depends on model parameters. Spin-fluctuation mediated
superconductivity is modeled by a Hubbard-Kanamori model with an effective
particle-particle interaction, calculated via the random phase approximation.
Magnetic orders are found using the Stoner criterion. The most likely
superconducting order we find has $d$-wave pairing, predominantly in the total
angular momentum, $J=1/2$ states. Moreover, an $s_{\pm}$-order which mixes
different bands is found at high Hund's coupling, and at high strain
anisotropic $s$- and $d$-wave orders emerge. Finally, we show that in a
fine-tuned region of parameters a spin-triplet $p$-wave order exists. The
combination of strong spin-orbit coupling, interactions, and a sensitivity of
the band structure to strain proves a fruitful avenue for engineering new
quantum phases.
