iPhone 4s does not start, it stays at the apple screen

Phone started malfunctioning, battery was draining quick, and it will turn off. sometimes it will let it charge, some other times it will not. got to charge it a few times. This one time, it stopped showing me if it was charging. the apple icon comes up, it stays for 1 minute or 2 and then the phone goes dark. I assumed it was the battery or the port, maybe poor connection.

So, I replaced the Battery, and power port. but the same thing, the apple logo comes up for a few minutes, and it goes dark (off). tried to do hard resets, erasing the phone thru the iCloud service, NOTHING... I am running out of options.... any other suggestions?? it will be appreciated.