Hey, so sorry to hear about your problem. I have actually ran into this problem a lot. 1 of 2 things should work. First try doing hard reset. Please be advised that doing a hard reset will completely delete all the information on the phone including contacts and pictures. To do a hard reset, you make sure the phone is completely off, then use a button combination to bring up the reset menu. For the note 4, you press and hold the volume up key, the homebutton and the power button until the phone starts to power up and has little blue writing at the top left of the screen above the galaxy logo. Once you see the blue writing, you can release the buttons. then use the volume keys to scroll up and down to factory data reset, then use the power button to select that option. then once you have confirmed that you are sure you want to do the reset, just sit the phone down and let it do its thing. I have seen a phone take over 4 hours to complete the process, but most of the time it gets done right away.
before preforming the hard reset, make sure you phone is fully charged, at least up to 80% and don't turn the phone off or reset it while its doing the reset.
If you do not want to loose your information, or you tried doing a hard reset and it did not work, then the issue is most likely your battery. Sometimes batteries get shorted out and cause it to make poor connection to the phone and forces the phone to shut down. Make sure where ever you get the new battery from, they have a return policy, just in case the phone just has a motherboard issue and you can return the battery so you don't lose any money.
I really hope this helped you out at least a little bit. Have a good day