my phone won't turn on. the apple icon turns on and then off

1. Restart the device. That's it.
2. Still not solved --> Force restart iPhone - Apple Support

1. Follow the Apple article to resolve the issue --> If your iPhone, iPad, or iPod touch won't turn on or is frozen ...
2. If not resolved; please follow the steps in the following link for support with this issue: If your iOS device restarts or displays the Apple logo or a spinning gear unexpectedly

• You still see a black screen after charging, force restarting, following the steps in this article.
• You see a black screen, but you can hear alerts, sounds, and vibrations.
• The screen remains black, but the device chirps when you unmute it and connect it to power.
• Your screen is on, but it doesn't respond when you tap, slide, or try other gestures. 
• You can't force your device to restart because a button is broken or stuck.
• Your device is stuck on the Apple logo, displays a solid color, or isn't recognized in iTunes.

Contact Apple Support to set up service if you have any of the above issues: