I saw this tour in Denver.
I recall it well. Bon Jovi were the up-and-coming young group while Ratt had seen their better days and were on the slope downwards.
I remember Bon Jovi had a banner hanging above the drums with their "Bon Jovi" logo, a total amateur job that looked like something the band members' girlfriends painted in the garage.
I recall they played a short 35-minute set and sounded like crap. Probably not their fault. I imagine Ratt's management told the soundman, "Make these punks sound like sh*t if you wanna get paid!"
And when Bon Jovi walked off stage, half the crowd left with them. they were there to see Bon Jovi. They didn't give a sh*t about Ratt.
I just looked up the date. September 11th, 1985.
I saw that tour one week to the day later on 9/18/1985 at Pershing Auditorium in Lincoln Nebraska. I vaguely recall the cheesy banner behind them now that you mention it. I was pretty disappointed with Ratt. That band had a ton of good songs and it was a real downer that they were so drunk and sloppy. I went to see them only it seemed like a waste of funds. Although my friends and I were pretty tuned up so I guess I probably still enjoyed it.