Have you tried to update the software? and restored the iPhone? From the description, it might be a hardware issue, not mobile network. I would seek helps from Apple, they might be able to diagnose that problem.
Have you tried pressing and holding both the home and sleep / wake button for 10 seconds? Does the force restart fix it? If not, trying to go into DFU mode. You do the same for Wp seconds only instead of releasing both button, release only the sleep button and keep holding the home button for 5 seconds. These times are exact. Too long or too short and it won't work right. It should become a black screen. Plug it in to a computer with iTunes. See what can be done about diagnosing it through iTunes. Maybe it'll need to have the os reinstalled.