Bad news to tell you it is the logic board, and it's drawing too much current. I'm still trying to work out how to repair it, I'd like to know what components need replacing if its possible too. Or else u can buy an extended battery case for your iphone.
what a good combination, have u opened up the shielding ? is there much corrosion on the PCB ? try to clean with IPA, and then put it in an ultrasonic cleaner cos a couple of cycles, then dry it out and heat it up. make sure u dont see any corrosion left on the board, if you do, resolder the contact points. perhaps also try it with a new battery, more work in repairing an if4 then an if3. good luck :) Update: normally i use distilled water in the ultrasonic cleaner, then i heat it up then i reflow it
you would have to check with your back case, is it a genuine one or the after market china copies, as some of them are faulty, you should take the audio flex from your original housing and switch it with the one in the housing you wish to use, and the home flex, u check for connectivity with a multimeter as your pressing it and depressing it, if its all good there, then it's your system connector that needs replacing. 3gs home flex can be used in a 3g, the pins are longer too.