Take the battery out, pull the sim card out for at least three minutes then put back together and reboot. Lastly even though I doubt this is the case make sure you have mobile networks checked under your wireless and networks. I somehow did that once.
Last time that happened to me, had to do a hard reset. When I would try to update, 4g would come on then go away when it was done with that. Called Verizon and they last resorted to that. That was 3 months agoand it has been fine since... just remember to back up your data.
You may have to call Verizon. When I recently worked at their call center, they can quickly check for any listed outages or known issues with their network, such as a temporary outage in a city and if there is an ETA on when it will be resolved.