Power light goes doesnt stay on just blue light and then off

Without looking at the monitor to confirm this, I suspect you have failing capacitors which power the HT circuit, simply, the power needed to light the display, is leaking to earth, capacitors have lost the ability to store the charge necessary to illuminate the screen, which is why it is working erratically.
You need to get this checked out by a TV technician.

Please check your RAM

It sounds like the power supply capacitors have failed.

the blue light comes on when cold and goes off when engine is hot enough to use full power.if it stays on,the temp sensor is faulty.

try connecting ac adapter alone without the laptop, check if ac adapter light will stay green or blue.
try another power outlet using the ac adapter alone without the laptop check if the ac adapter light will stay as green or blue. if all fails replace ac adapter.

if ac adapter stays green or blue try connecting to the laptop without the batt, check if the ac adapter light will stay on, if yes then try connecting ac adapter to laptop with the batt plugin check if the light stays green. if not update the BIOS. if fails replace motherboard.

probably a blown capacitor open it up and look at the capacitor see if any one of them are curvy on the top blowend

Call Sony so they can fix it. This sounds like multiple problems. Good luck.

most notebook adapters i've seen have the light stay on as long as it's supplying power to the notebook. so your p/s maybe defective - my guess is that something's tripping it - are you sure the p/s is suitable for your particular notebook (based on voltage/current/power requirements)?

have good tec replace if nessary, The inverter module