What type of battery are you using? Do you have the right battery setting?
The camera's battery setting is set to "Coolpix" as the default (the original batteries it came with). If you're using regular alkalines, be sure to change this setting in the camera (Menu > set up > battery type > alkaline).
When the batteries ran out for the first time, I bought cheap dollar store batteries, and when I put them in my camera, it shut off immediately when I tried to turn it on, despite them being brand new. I found out this was because the batteries didn't have enough juice to power my camera. I replaced them with Duracell, and it worked fine after that (Panasonic, another dollar store battery works okay too).
Hope that helps.