It is possible the display is the problem, but if you are seeing the charging icon, that is less likely.
Try the following and see if it will turn on.
1. Remove the battery.
2. Press and hold the "Power" button for 20 seconds.
3. Now put the battery back in.
4. Plug it in to charge and allow it to fully charge before turning it on.
If this doesn't help, then test the charger cord to see if it is good. Either try your cord in another device, or try another devices cord on your device.
Should that not be the problem, then check the battery to see if it needs replacing.
If you still can't get it working, and if it isn't under warranty, then I would try a local tech.
NOTE: Yes, I know some phone batteries are not so easy to get to, on some devices. In such a case, make sure that your phone is not under warranty still before you try opening it up yourself. If it is under warranty, then contact the manufacturer first.