Do I replace the logic board or the battery?

My PSP go has a rather curious problem. When I first got it pre-owned it worked really well. But a few months later it shut off and would not charge or turn on. Eventually I got it to charge a few weeks later, but then it occurred again. Naturally I assumed it was the battery so I carefully removed the battery, and attempted to turn it on again (while the battery was out). To my dismay it did not turn on when plugged into the charger. I ordered a new charger as suggested by someone online, but that did not help. However, when I did the same procedure again (removing the battery and all) my PSP go magically came back to life so I just assumed I had knocked the battery out of place. I fully charged it, and it was working great until I left my house with it and it stopped working again. I am not sure if I should replace the battery or what. If someone could help me to better understand my problem that would be most appreciated.