I have a Dell 2407WFPb monitor that will only stay on for 3 to 4 seconds at boot up and then shuts off. Warranty has run out. No signs of anything wrong until it just went out. I can turn it off then back on and it will stay on for another few seconds then turn off. When the screen goes black the green power button light stays on.
Sounds like a power supply problem in the monitor. I don't know if you can get it fixed. Dell doesn't do out of warranty repairs and it may be hard or even impossible to find a place that can legitimately repair it. Sadly, the best move is to look for a new monitor.
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My (Non-Dell) monitor has been doing that too, and my Inspiron is only months old. The kid can play online vid games all day, but when I sign into my account and do the usual web surf, the monitor just shuts down. It works for awhile if I run a comprehensive Virus Program, then it starts cutting off again like someone pulled the plug. I don't believe it's a monitor problem. It doesn't shut off if I leave it on alone and power off the desktop.