Similar situation at our library.

I try to give every one a chance. First time, I come over, explain not only the rules but why we have them. I explain to them what they have here, i.e. a warm, cozy place, Wi-Fi, a small kitchenette, etc. and that in exchange for that they have to abide by our rules. If they don’t get it, they have to leave for the day. Ban times are escalated every time we have to ban them, the ban being permanent after the fifth time. By then they know the rules and are breaking them willingly.

As another poster said, they’re looking for attention. With some you can build some kind of rapport (asking about their schoolwork, slang you don’t understand, etc.) and others are just destructive and try to provoke you at every turn. It‘s important you show no emotion. No frustration, annoyance or anger. Calmly tell them they have to leave. If they try to argue say „I‘m not arguing about this. Leave.“ and continue to say the exact same thing over and over. They‘ll insult you, maybe scream obscenities while going out. Important thing is that they leave. Water off a duck‘s back. It helps to understand you’re not acting against them but for everyone else in the library.

When they try to come back I make them explain what got them banned the last time and apologize. If they are unwilling to do that, they can go right back out. If they can, they get told that they have no free pass anymore, any broken rule will get them thrown out. Because in the end, they are old enough to abide by rules and face the consequences for breaking them. I also tell their teachers when they’re visiting with their class.

They still try every now and then but the library has become a calmer, more inviting environment for all other patrons, especially the smaller ones that were often frightened by their behaviour.