I'm having troubles with my phone, my cousin gave me the phone a couple days ago after he upgraded to the HTC Sensation.
He was just gonna throw away the Galaxy S 4g, until I asked him for it, it's a little messed up, needs the lcd replaced (no major damage, just has a couple of very minor cracks in the glass.)
So he told me it had a problem turning on, what happens is when you try to hold down the power button, it won't turn on, it will just VERY briefly show a battery image with a circle with many spokes on it, then shut right back off.
this happens even if you try to do a hard reset and such.
However when you plug it in to AC power and then turn it on, it works without problem, here is where it gets weird though. When you unplug it, it continues to work fine, and doesn't shut itself off.
I'm at a loss, i'm willing to replace parts myself, i'm fairly confortable taking apart electronics as I work in a PC Repair store, and fix Xbox 360's on the side.