MM has a point. I often use rechargeable batteries but sometimes an item will not accept they are providing enough power, for reasons I am unsure of, so I use new non-chargable ones and all is then ok.
I appreciate it is of limited help, but if I were sure the batteries were ok and installed correctly, and the camera refused to stay on, I'd conclude it had some kind of fault that could only be fixed at the workshop. (And probably easier to replace.) Is it still under warranty ?
Aside from that you could look for broken connections maybe. Since it does come on I guess the batteries are providing some power, but for some reason apparently not enough to satisfy the camera electronics.