Test it with the battery out, try one stick of RAM at a time, then try and reset the processor , test it to an external monitor that's the testing i do. if they fail then you know you have a logic board problem.
try resetting the digitizer on the logic board ifixit has a guide on how to do it, if this does not fix your issue then you might want to consider a new display. it a may also be a board issue. happy repairing
connect it to itunes and do a factory restore on it, it may be software corruption, if not strip the device and use WD-40 contact cleaner if this doesnt solve it there may be a chance its the logic board
press and hold the home and on/off button until the apple logo appears then release your finger from the on/off button while keeping your finger on the home button until the connect to itunes screen appears. "it might take a few goes to get this right" happy repairing