Replace the digitizer. Aka the touch screen

Maybe time for a new phone.

Check the LCD connections. I usually don't remove the screen when doing a replacement.

If that sounds like too much too do, take it to a repair shop.

LCD is bad. Needs to be replaced.

I am going to try and replace the baseband chip tomorrow. We will see. I will let you all know Update (05/19/2016): It was the baseband chip. Heated it up and phone restored. The solder points became loose from over charging and heat.

Or clean it with isotopic alcohol carefully just the button. Let it dry. The. A little dust off. There nitorious for getting stuck

Try disconnecting the battery there is a small wire circular antenna that plugs in near the battery connection make sure that is connected and that the screws that go back on with the battery put her in the right positions after your reconnect the battery It will say searching after your boot up wait for a minute or so and then do a hard reset holding the home and power button and you should be good to go. If it doesn't get fixed you need to replace the dock connector which is one of the antennas

Make sure that your flex cable on your screen on the backside where it connects to the LCD and digitizer or are firmly pressed down and that the back metal plate is not over the long metal plate. Try unplugging the battery I have had this problem after he screen replacement And then it worked fine. Also if one camera goes out both will not work this is common.

Sounds like overheating to me. Maybe the copper coil is in grounding correctly to the chip. What size display?