Have you tried doing a soft or hard reset on the phone? Have you taken the phone apart and checked the connections to the logic board? If not I would try to do a soft reset by either taking the battery out of the phone for 30 seconds and putting it back in and powering up, or hold the power button down until the phone turns off then try turning it back on. If this does nothing try the hard reset. You will loosed your saved data doing the hard reset and the phone will be set back to factory settings. If you do mind loosing your saved data then do the hard reset by holding the power button + volume up button + home button, release all buttons if and when the Android logo appears. If you don't want to reset your phone to factory settings then start checking for hardware issues.
If resetting or restarting the device solves nothing then I would troubleshoot the LCD next. To me it sounds like an issue with the LCD and I would open the phone to access the logic board. Then I would check the LCD connection to the logic board to make sure it is seated correctly and not damaged. If your phone has had any exposure to liquid I would clean the logic board and connections with Isopropyl alcohol (90%-99%) and a small brush such as a tooth brush and this will help remove any corrosion that may have built up and caused a poor connection. Also make sure to look over the LCDs ribbon cable well because it may have a tear somewhere which will cause it to fail. If the phone has been dropped the LCD may have been damaged or its cable or connection.
Best of Luck.