Verified answer

you got a lemon....trade it in for a more reliable one.

Might be a fuel pump problem. The fuel tank is pressurized by fuel vapors, but when you open the cap, you release the pressure. The pressure might be helping out a fuel pump that isn't working right, and when you release the pressure, the fuel pressure drops until the pump can catch up again. Or something like that. That's my quasi-car-guy hypothesis. Try giving the fuel pump more time by pausing with the key at ON for a few seconds, or a minute, and then starting the engine.

You should ask this question at ElantraClub.

Or it could be the fuel filter. Gum that thing up and nothing gets to the pump. It's a cheap fix, the filter's about 20 bucks and is located under the back seat. You can do it if you're handy with a wrench, or pay 100+ bucks for someone else to do it.