My iPhone 3GS shuts off by itself. What should I do?

So when I unlock the screen from the slide lock it is on but usually only or a minute. If I play or game or record a video or go to the Internet and sometimes when I'm texting to basically when I do anything that keeps the phone on for more than a few seconds it shuts off. When I try to turn it on it doesn't work and then when I try to reboot with the home and power button it brings up the logo but usually shuts off while it's loading. Sometimes it turns on again. But when I plug it in it turns on. Even when I'm on the phone in a call it shuts off. Ive tried restoring it and that didn't work and I have to try to reboot it everytime. Is there anything I could do besides those that have worked for you if it has happened? I've read other articles and forums but they just say to reboot but that doesn't work because it will just do it again.
 
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