Alot of HP and Dell monitors run into this problem even though other brands of monitors as well can have this issue.
What happens in most cases is that there are puffy capacitors on either the power board or main board inside of the monitor itself. What needs to happen is the base of the monitor needs to be taken off, then the screws taken out of the back. Take a flathead screwdriver and there should be grooves all around the edges of the monitor, put the flathead inside of these grooves and twist the screwdriver and it should unsnap the front bezel from the back bezel.
You might have to do this all around the monitor to get enough of it loose so you can pull it all off. Once you get the back off then you should see a metal shielding like a metal box. Go ahead and unscrew the screws holding that in.
Once that is taken off, you'll see two boards, a power board where you would normally hook up a plug to it and a main board next to it which is bigger in size.
What we are looking at is the capacitors, a good capacitor looks like this:
Notice the good flat silver tops.
A bad capacitor looks like this:
You'll notice that the silver tops are swollen. If you see these then they need to be replaced. You need to have soldering experience cause the old ones will need to be taken out and the news ones put in.
Make sure the ones you put in are of the same UF rating and at least the same voltage or higher. Make sure before doing any work on the monitor that its not plugged in.
Once the new capacitors are put in then go ahead and try turning the monitor on. If it turns on then BOOM your good to go. If if still doesn't turn on then its either a bad board or the screen itself is bad but either case the monitor would need to be replaced.