Hello there,
There are at least 2 possible problems here.
.- You powersupply is somehow not working properly, I don't think this is the case since it starts then it goes down, but as a bet you can give it a try, just get another ATX powersupply,borrow it from a friend of your and test it.
.- The second one (which I think is the one) your ATX power on button on your mainboard is not ok or your mainboard is not ok. To find out if it is the problem just open your computer, look for the poweron button and make a short circuit on it, you can use a jumper, just make a short on it and remove it. If the system starts up it means that your power on located on the case is not ok, just fix it and it will work.
If the mainboard doesn't start up it means that your mainboard is not ok anymore.
I hope it helps, please e-mail me back if it doesn't work.
Regards,
@lberto
alberto@pancarib.com