Should be easy enough to check... use a digital volt meter and backprobe the 3.3,5.0,12.0 rails while the computer is trying to boot or stay running.you should see 1 or all the rails dipping real low if its power supply.if you dont have a meter go to radio shack and buy an inexpensive one for 20-30 dollars or borrow one from a friend.its nice to have one around when you have computers.sounds from your description it may be a power supply but it is within the realm of possibility it could be a motherboard.another option is to swap a p/supply from another computer.as a last resort unplug all unnecessary components in your case and see if it changes the way it runs.if its power supply the fewer components it has to power up may give it more to boot with and it may boot up fine.then you know its a power supply.the best way is to check it with a volt meter.its not a bad investment as you may from time to time have to check another computer and have to check voltages.also its nice to check what your software reads as voltages vs what the voltmeter says the voltage is.
my bios and software read my 12v line as 11.77v but my volt meter reads my 12v rail as 12.23 steady ,even at boot up it only drops to 12.18v.buy one tomorrow and you will have your answer.....................