What is Shounen Bat? Who or what is Shounen Bat?  


   Does Shounen Bat exist in the mind?  Or does he exists in reality to relieve the pain of the people who are cornered by reality?  


It became unclear to me who or what Shounen Bat was after


   the copycat died.
There's no explicit explanation on who or what he is.


  Does Shounen Bat exist in the mind?


Not so much in the imagination, as other people can obviously see Lil' Slugger. It's possible that he could exist in the minds of everyone. Whether or not he exists in reality is a matter of debate; when he attacks people, is it physical or are his victims simply manifesting the injury?

Regardless, 


   it seems he's been manifested by Tsukiko, the character designer and first "victim". Except she reveals she was never attacked by Lil' Slugger and that Lil' Slugger was her manifestation when she was a child to take the blame of all the bad things that she did, like letting her dog die. She eventually believed the delusion and somehow the delusion manifested into either reality or within the minds of everyone, growing in strength from people believing in his existence.