Christmas this year? Then it's in warranty. Bring it in to Verizon for an exchange. An older one, out of warranty? See if you can get into download mode (Volume Down/Home/Power). If you can, download
Odin (to your PC) and the ROM for that model S3 from
SamMobile Firmwares (make sure it's the right country/carrier) and flash the ROM.
If it can't get into download mode, you'll need a Samsung Dongle. (You can get ripped off for about a buck including shipping for one on eBay.) If there are no instructions, search
samsung dongle and you'll find a lot of sites with instructions. (It's trivial.) Those things are used to bet totally bricked Samsungs into download mode. (Totally bricked means that the phone reacts about the same way a brick does - nothing at all, not even a quick flash of the back button.)
If even the dongle doesn't work, most likely the motherboard is fried. It would probably be a bit cheaper to have it replaced than to buy a new S3.