Switch on. Runs for a few seconds and switches off again ?

If my computer has been switched off for more than a few hours, I struggle to restart it. Fan starts for a few seconds, then powers down again. After many attempts, gets going. If I leave it on, but with power options to switch off display and h-drive only, can keep it OK for months. But next time completely switched off for more that an hour or so, I get the same problem. Someone suggested the power switch may be the problem. Any ideas ?