Definitely sounds like a PSU issue. And unfortunately Alienwares use a proprietary PSU, which has some special lead wires for the lights and other Alienware special perks/apps so you cant just order a Corsair unit from Newegg. They are also pricey for what you get.
You can find them though, so I would start there - try getting one of the 875 watt Alienware units and replacing it. I'm about 99% sure this will fix your problem. Take your time and you can replace it yourself.
I would of course first double check all of the connections from the PSU if you had it installed by some repairman. He could have left something unplugged, or not completely plugged something in.