Not sure if this response is outdated, but try this:
Take out the battery, then press and hold the ON button for 7-10 seconds (the longer the better). Reinsert the battery, then press ON button to start it. This is a soft reset, and wipes the temporary memory and empties all remaining power in the circuits. This helps on many phones and even laptops that won't start properly or show anything on screen.
If this does not work, try the next step. Again, do as above. But after inserting the battery, do not turn the phone on, but press and hold the ON button and the HOME button at the same time for 7-10 seconds. This is a hard reset, and works best if done after a soft reset. Just be sure the battery has at least some juice in it and isn't empty. If possible, try with another battery if this does not work.
If nothing works, try to have it swapped on DOA or if not appliccable, have it sent in to be fixed.
Hope this helps!
P.S.:
Yeah that doesn't sound very good. be cheeky when you take it back and ask for a case or some refund for your'e troubles that works sometimes.
It's not the fault of the store or any of its employees that you were unlucky enough to get a faulty device. I don't know of any store at all that opens and checks that every product in their store works. The store is not responsible to compensate for any "troubles" you may have suffered. When you purchase electronics, you know and understand that not all products always work 100% perfectly, because in mass production, nothing's produced 100% perfect. This is a universal fact everyone is, and should be, highly aware of. But, you are entitled to a swap unit if you have the whole box and all contents of it (this also depends on the current warranty laws in your country, but generally they're globally the same), and if they have no swap unit, you may claim to have your money back if they have no unit there at that time.
