Brief summary of hand

Dealt KK UTG+2 and raised 4xBB (very tight table and this normally folded all but one or two players) - all fold but small blind calls and big blind calls.

Flop comes J/4/9 rainbow and I bet 6xBB. Small blind folds and BB reraises me another 6xBB. I'm thinking maybe J/9 and he hit 2 pair but he hopefully wouldn't call a 4xBB raise with that preflop. Also considered JJ. I call. 4th street is a Q. I bet and he raises me again.

By now I'm a little scared. My read on him was that he was LAG so I didn't put him on a really good hand. This is where I considered laying it down thinking he may have had JJ, QQ or AA. Would you lay this down? Or any advice besides lay it down?

Results were that I went all in and he had AA and I lost my buyin. But I look at it and think I didn't play anything wrong except to listen to my little voice saying he has AA.