This post is to be read when all the other options have been exhausted and you are ready to tear apart the phone.
First. BUY THE iFixit KIT to work on your phone. They do a lot of good.
Second, bring up the phone tear down. Study it, and get ready to take it apart.
One of the Issues I have seen has been the actull soder joint to the battery being broken. This is due to the battery floating around in the case. SO get a new battery ready. OR you can fix the solder on the battery board.
But that is how I fixed my sons phone. The Actual battery installation was Culprit from the previous owner.
The other issue is often the On Off Switch that is on the side. The plastic/Circuit board goes faulty. Its a bear to replace, but can be done. The part is only a $10.00 at most.
I hope this helps you out. GOod Luck!
iMickey503