If your iPhone will not turn on, even though you think it is fully charged the first thing you should do is try to reset it by holding the Home and Sleep/Wake buttons down at the same time and continue holding them for at least thirty seconds or until you see the Apple Logo. If resetting your iPhone doesn’t turn it on, then you might need to replace the battery. If your phone is still under warranty, this should be done free of charge, otherwise you may have to pay for a new battery and to have it installed in your iPhone.