First, if you have not already done so, remove the memory card and attempt to power the camera up without it inserted. If the camera powers up or responds without the card please let me know.
Next,to make sure that poor contact is not causing the issue, since you are positive that the batteries start out good. Remove them from the camera and clean the metal battery contacts inside of the battery compartment and on the battery with a q-tip and alcohol. The connection is very critical and a very small amount of residue or dust can inhibit the power transfer. Also, check how well the battery door shuts and if necessary adjust the metal contacts by gently prying them with a small screw driver to increase the tension.