Hi i have the exact same problem. Vita3g completely charged and powered off in airplane mode.
After 1 day power it up its lost 10 percent charge.
After 3 days power it up only 20 percent left
After 5 days its dead.
I changed the battery, no change
I changed the motherboard, no change
Now i left out all the 3g bits and its holding the same charge (50 percent) for the past 3 days.
I ordered a replacement 3g modem from ebay (france) and with the new 3g modem it does not drain anymore.
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I also have to reset the date/time every time I power it up after the battery is depleted although all of my other settings remain intact. This is also the original model with an OLED screen.
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