Won't boot after battery replacement

Hi all,

two years ago my MBP was diagnosed "dead" because of oxidized motherboard (a terrible accident including me, a glass of water and... my cat).

So, last week I tried to boot up after 2 years, and it boots up ! (seems to work well)

But the battery was bloating, so I decided to replace it.

Now my problem is :

- If AC adpater plugged but no battery connected = Boot

- If I connect the battery + plug AC adapter = Boot only the first time

- I can "restart" my computer, I can't "shutdown" + press power button

- When my computer is running, I can plug off AC adpter, no problem (I can see charging percentage on system tray)

- I can also see charging percentage with leds on the computer side

What can I do, any Ideas ?