Could be simple or major hardware failure, including Water Damage.
Clean iPhone Dock Port with a clean dry toothbush.
Plug iPhone in with Wall Charger for 10 to 15 minutes. While still plugged in Reset, hold both Home and Power buttons until the iPhone restarts. This can be difficult to determine when Apple Logo is ON and OFF. Try Reset like this a couple of times like 2 minutes apart.
If still problem, Restore Firmware, connect iPhone to computer with iTunes open, hold both Home and Power buttons until the iPhone is Recognized in Recovery Mode, usually about 20-30 seconds, Restore the Firmware.
If still problem, try DFU Mode Restore.
If still problem and you have Warranty, which you should with Verizon iPhone, return to Apple for resolution.