iPhone doesn't charge when battery completely drained

Hi - I have an iPhone 5 that doesn't recharge when the battery is completely dead (when the screen is black and won't even show the low battery sign). I've replaced my battery 3-4 times and every time the new battery works, but within a few weeks when the phone dies because of low battery, it refuses to charge and turn on and the only so far has been replacing the battery. One the battery is replaced, the phone works perfectly fine, it's only once it's completely drained that it refuses to charge again.