The only trouble I can recall reading about is in an interview some years ago in a swedish magazine (I can´t remember where or when right now).
There he mentioned that his father "Stellan Skarsgård" had to pic him up from the
policestation one night or morning because he had been in a fight that night in a bar in the neighbourhood. He was younger then. I got the impression from the interview that it might have happened before.
He also said in the interview that his father "had a talk with him" where Stellan said that Alex should think about what he was doing and that he was to smart to be doing this stupid things.
Those early Swedish interviews whith him (often in magazines for men) are rather interesting to read because he was much more open and easy to talk with.
I suppose that is the downside of fame that he no longer can talk so freely.
I think he got busted very being very publically drunk or something similar, but that was back in his teens if my memory is correct.
Which is one of the reasons I think Cindy Adams just cobbled together something from older interviews just to mention him in the column.