Chances are your car is not charging properly. Be sure that the red CHARGE light comes on when you turn on ignition, but goes off once car starts. These cars have a very complex engine computer that requires almost a full 12 volts to run correctly. Unless your car has the Bose option, the factory radio should not draw enough power to affect the engine if the battery had enough power to start the car period.
A good test that the car is charging at all is to shut off all accessories but leave engine running while you CAREFULLY disconnect your negative battery cable. Its best to wear eye protection just in case, because there will be some spark. If the engine stops right away, the car is not charging enough, or not at all and you more than likely need an alternator.