Desktop powers on and off dozens of times before it stays on

Hello,

When I touch the power button on the front of the bezel of a no-name desktop computer, the fans and drives power up for one or two seconds and then go dead for a few seconds before trying again with the same results. If you let this continue for several minutes or sometimes hours, it usually finally boots up and operates normally until you shut it down.

I suspect a problem with the power supply, but I'm wondering if anyone else has experienced the same behaviour.