could be from a blown VRM (voltage-regulating-module-next to the GPU)
First things first. Try it on a PC , next>
Clean it well , use dry contact cleaner.
Put new special GPU paste (not to much)
Test. (preferably on PC first)
if you wish , pre-heat the card with a hair dryer (wont do more damidge at 50-75cm far).
if you have a volt meter, try getting schismatics from online and measuring stuff on it.
If all else fails , can try cleaning with WD40+brake disk cleaner (but as last restort)
P.S. Not impossible to repair , but hard to identify exact problem.