I'm actually having the same problem. It just started his morning.
The phone won't boot normally. I get the kernel boot screen, sometimes a Samsung screen, and then it turns off. 2-3 seconds max.
I can get it into download mode, but after 3-4 seconds, it spontaneously shuts off.
Recovery mode (CWM) is completely stable. I can get in, make back ups, perform a key test, etc. When I run a key test, it shows that the power button is a little sensitive, but not crazy. If I jiggle it some (mostly back a little) it registers a key press, even if I'm not fully pressing it. If I do press it fully, sometimes it doesn't register.
If this is really a button issue, why is recovery stable? Even if I put the phone on a flat surface with nothing that could touch the button, the phone still doesn't stay in download mode (it just shuts off.) When I put the phone in recovery, place it on a flat surface, and perform the key test, the phone isn't registering a key touch.
I've tried 3 different batteries, kept the phone plugged in, popped the battery and held the power button to discharge everything, wiped the cache, wiped dalvik cache, and used CWM touch to reboot (and avoid the power button all together.)
The only think I can think of that's maybe different is that gapps was FC'ing more than normal this morning (before the phone crapped out.)
I'm running StockMOD ROM 4.4.2 R2 with the Ktoonsez Kernel for TouchWiz 4.4.2.
EDIT: Also, when it's plugged in, the LED doesn't light up. I'm not sure that it's actually charging. I do have a wall charger, so I can verify that the batteries I'm using are charged.