I have the same problem with my son's phone. We would charge it fully and it would shut down if it was jostled in any way. It would only come back on after the charging. It even shut down while we were unplugging the charger from it. I took it to the Verizon store and it works fine with a new battery. Try plugging it in to a portable battery and see if it works. Batteries are pretty cheap online, but considering using the factory warranty since it's only 6 months old now that I've seen your post.
If throwing a new battery at the problem solves it, then I think the issue may be a battery that has changed it's dimensions after repeated charge / discharge cycles, and not fitting fully into the battery compartment and making a good connection with the phone. A new battery might be the answer, or re-seating your own battery.
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