Unfortunately, the issues you are experiencing are symptomatic of an internal hardware defect, meaning a faulty power supply or main/control board, though there are many possible failure points. This issue could be a manufacturing defect, but is also often caused by a power event (shock/surge, prolonged over/under voltage condition, etc).
This could also be the result of age related wear and tear; as a 2011 model tv, that type of failure wouldn’t be surprising. This tv is at/past its service life at this point.
There is one step you can take here, both as a possible fix, as well as a diagnostic evaluation:
1) Unplug the power cord from the outlet
2) Wait at least 1 hour (60+ Minutes)
3) Press and hold the power button (on the set, not the remote) for 60 seconds
4) Keep holding power button while plugging back in, and for another 60 seconds after. When plugging back in, go directly to the wall outlet, bypassing any power strips or surge protectors.
If this reset process does not work, it would unfortunately confirm an internal hardware failure.
This is something that would require repairs to correct; if you're thinking about a do it yourself repair, you'll need to be comfortable and skilled with electronics repairs, which could include dismantling the set (removing the back panel of the tv), diagnostics testing with a multimeter, changing out boards as needed, and possibly desoldering/removing and soldering/replacing individual components.
If you’re considering repair at a local shop, you’ll find a typical repair to be around $300 for diagnostics, parts, and labor.
For a tv of this size/age however, replacement will make more sense. You can get a brand new LED tv with a fresh warranty in the $150 range.
I can help you with finding parts, finding a repair shop, or recommending replacement options, if desired.
My apologies for the news,