iPhone has charging issues, what would need to be replaced?

Hello, long time lurker, first time poster.

I have recently run into troubles with my iPhone 3gs. When I plug it into either the usb or wall charger it refuses to charge. It only charges when I turn an app on, and let it sit for a bit (but I have to leave the app on).

I've already purchased several batteries (about 3 in total), and I've ordered and replaced the docking connector all with no luck. Any suggestions on what I should replace next or look for would be awesome. I'm thinking the logic board, but I'm not sure.

Thanks in advance!