If you have an Apple Store in your area you should take the phone to them so they can check it out. It could be a bad cord, charger, or the device. You should still be in the one year manufacturer warranty (although if you got it at release your year is almost up assuming you don't have Apple Care Plus). If you're not near a store you can contact Apple for support over the phone (or online). Your other option is visiting a VZW for help.
Alexann, that frequent restarting can be very irritating. Have no fear we are here to help you. Have you tried any other chargers? How long has this been happening? Is your software up to date? These things can help, please let us know so we can further assist you.
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I tried other chargers with no luck. I have only had my phone for 6 days. All iOS updates are in. This is the 1st time the phone has went dead while on, usually I charge when it is low.