Bobbi Kristina Brown -- Police Report Filed in Neighbor War

Cops are now involved in Bobbi Kristina's nasty feud with her former apartment neighbor -- TMZ has learned the guy on the receiving end of BK's bitchy tell-off letter has filed a police report.

We've learned ... after Josh Morse discovered a hate letter on his door yesterday, calling him and his wife a "miserable couple ... sh*t at the bottom of our shoe" ... he went straight to the Alpharetta Police Department in Alpharetta, GA.

According to the police report, obtained by TMZ ... Josh wanted the incident documented "due to the ongoing issues he has been having with the couple."

As we previously reported, Morse and his wife had filed more than 10 noise complaints against Bobbi and her BF Nick Gordon -- claiming they were the neighbors from hell.

BK responded with a letter blasting the couple as horrible people who were just jealous of BK's life.

We're told ... BK and NG are NOT being investigated for the incident because there are no allegations of criminal acts.