I had the same problem with a laptop.
It only powers up when the battery chager is plugged in,although it works from battery after starting up so the battery is not flat.Try the same.
(Maybe its the battery can`t supply the higher current peak when switching on).
I have an older Dell notebook, it has been sitting idle for about 18 monthsand now wont power up at all. Any suggestions? I was thinking that maybe the internal battey might have gone flat and not allow startup or something weird like that. All thoughts welcome.