I have a similar issue: if i live my Mac mini 2011 off for a night, it will boot and start sierra, and then, after a few minutes computer will shut off by itself, and if i try to turn it on again, all i get is the cooler at full speed and no screen at all (not white not black not grey no logo, nothing). At this stage only fans work, no other noise of booting, forcing me to shut down computer by holding power button. After a few ours unplugged, it will star again, and same problem occurs, finally shutting down computer and secondary problem start (fans at full speed) . Any help would be very appreciated.