Eight seconds after powering on, my phone goes dark and becomes unresponsive. What can I do? I have a Samsung Galaxy S Relay. (This is kind of like a Samsung Galaxy S III plus a slide-out physical keyboard.) It's running stock Android 4.1.2, rooted using TWRP and SuperSU.

I was using Spotify yesterday when things started to go wrong. Soon, I saw only blackness and the phone's buttons stopped working — even the power button was useless. Also, when I connected my phone to my computer using a USB cable, the computer's lsusb command reported that there were no USB devices plugged in, and the dmesg command indicated that there had been nary a peep from the phone. Maybe the phone was off.

I had no choice but to remove and reinsert the phone's battery.

When I reinsert the battery, the phone seems to start working again, but only for eight seconds. I can take advantage of these eight seconds in different ways:


I can attempt a normal boot. I see the "Samsung" logo screen, then the Galaxy S Relay logo screen, then everything goes dark.
If I press certain buttons, I can enter Download Mode very briefly before the screen goes dark.
If I press other buttons, I can ask the phone to enter recovery mode. I see "Samsung" and "Recovery booting...." before everything goes dark.
If I connect the charging cable and the battery at the same time, I see the usual graphic telling me to wait for the charging status display software to load and start itself. But the screen goes dark, and the charging status display never appears.


Once the screen goes dark, it stays dark. Even if I hold the power button down for 30 seconds, then let go for a bit, then hold it down for another 30 seconds, nothing happens. If I want to make the phone work for another eight seconds, I must first remove and reinsert the battery.

I don't think the battery is empty. My multimeter says that the battery is at about 3.9 volts. For a lithium-ion battery, 3.9 volts means "90% full".



What troubleshooting steps can I try next?

[Edit: This post is basically a verbatim copy of a Reddit post of mine.]
I have owned several Samsung galaxy devices (including an S3 which is somewhat similar to your phone). Often I find that the battery becomes faulty after 2 years of usage. The symptoms you mention also sounds like battery related problems. The key giveaway is that you unable to sustain download mode before your screen goes dark. 

Troubleshooting


Try replacing the battery. This will eliminate the possibility of a
faulty battery. After replacing the battery, see whether your phone
will stay on for more than a few seconds.
If you unable to acquire a second battery,  try booting your phone whilst in the charger. It should be noted that using the phone
whilst charging isn't usually a good idea, however for
troubleshooting it should be fine. 
Check your charger, wires should not be frayed or worn. Worn
chargers can affect your phone's battery.