I have one of the original PT111 models. No problems with mine, though admittedly, I haven't shot it much these past 2-3 years. Obviously it's a DAO. It definitely has a spongy trigger. I could shoot it well, but only if I shot it alone. I always had to stage the trigger. By that I mean I would pull the loooooong trigger to a point just before it would break, make my final sight and then pull through the final fraction of an inch to cause it to fire. This took a little longer than you might take with a single action trigger. That said, the thing was rock solid. I still keep it in one of my stash areas. (smile) I replaced it with a MilPro 9 mm basically because of the nicer trigger.
As for known problems, I'm going back a few years now, but as I recall, some of the original PT111s self destructed when the frame pins backed out. It didn't appear that this was a major issue for most pistols, but there was at least one example which was broadcast around the web. This was different from the problem with the original PT145s which developed cracks in the rear of the polymer frame. Again, both problems were reported with the EARLY original PT111s and PT145s. I haven't heard a report of a frame crack on a PT145 in years. Any issue with the PT111s also appear to have been corrected with the later models. There never were many reports of disintegrating PT111s.