Can You Play Mario Kart on PS3? No—Nintendo Exclusive

No. Mario Kart is a Nintendo first-party franchise developed and published exclusively for Nintendo consoles. The PlayStation 3 is a Sony platform. There is no official version, port, or equivalent of Mario Kart on PS3.

Kart Racing Alternatives on PS3

• ModNation Racers — Deep track/character/kart creation, solid handling.
• LittleBigPlanet Karting — Creative, physics-based, UGC-focused.
• Sonic & All-Stars Racing Transformed — Excellent transformation mechanics (land/air/water), great Sega roster.
• Crash Team Racing Nitro-Fueled — Remaster of the PS1 classic; tight drifting, strong nostalgia.

If you want Mario Kart, you need a Nintendo system (Wii, Wii U, Switch, 3DS, etc.).