iPhone 4 will not work without charger?

Hi folks.

I have an iPhone 4 here that behaves rather funny.

It only starts with charger connected (wall, not computer). The battery shows 100%, and the phone tells me to connect to iTunes.

Since it dies upon disconnecting the charger, I thought it might be an issue with the battery, so I switched to another that I know is working. Still the same behavior. I changed the charging port just to see what happened, still no avail.

I'd say this is a logicboard issue, would anyone agree?

The new battery was NOT fully charged, still it said 100%. DFU not possible since it wont start with USB-charger.

Does anyone have any ideas where to start looking for the problem (not in a dunk-college-douche kind of way)?

Sincerely

Me