Two things to try before the third (although I'm leaning toward the third here)
Try cycling the display by pressing Fn+F7 keys also try the Display Off key
See if any display in BIOS. To access BIOS, rapidly tap the delete key after starting the computer (you can't tap it too much) if no display then, see below - if you do get a display in BIOS let us know
If the above doesn't work, I'm going to recommend looking into having the GPU "reballed", it is a specialty process that few shops have the ability to perform but it seems to be what's needed. It may be a little expensive and depending upon your location, the laptop may need to be sent off to have the service performed. Another service that may work is known as reflowing but I won't recommend that since it's usually a very temporary (if successful at all) fix - I've personally only seen one graphics card last an entire year after a reflow (I'd say the average is a few months). While reflow is an option, I would not recommend it here
Although it shouldn't need to be said, there are some very flaky companies out there, I'll suggest researching the company thoroughly before sending off your laptop (if that's what's needed) for repair, call or talk to them personally, ask questions to get a feel for their customer service level also - not everyone is a computer guru, don't be afraid to ask "stupid" questions