1. Have you tried taking the battery out for a minute, replacing it, then powering on?
2. Are you sure the battery is well charged? Take the battery out, replace it, and then plug in the charger (without pressing the Power button) - after a moment you should see a battery graphic on the screen showing the level of charge.
3. Try booting into Recovery mode - with the phone completely off (ala battery pull), press and hold the Volume Up and Home and Power buttons simultaneously, and release them all when you see the Samsung splash screen.
If this takes you to the Recovery menu, select "Clear cache" (use the Volume rockers to navigate the UI and the Power button to select), then restart.
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