We are all individual so there is no universal mattress recommendation that's good for everyone of us. It depends a lot on what back problems you have, what sleeping position you use, etc. And somebody with identical problems might still feel better on a different mattress than you. It all comes down to "try before you buy" in my experience. I know that's not easy with a mattress, but I noticed differences when I slept at hotels and used that to decide on my own mattress. For instance, I had a hard mattress at home, which I believed was good for me, but when I went overseas and slept in a softer hotel bed, my back pain actually got better. A few years prior to that, I was convinced that whenever I sleep in a softer mattress things get worse, and it was probably true at that time. I also experimented a lot with pillows and sleeping positions, and I still happen to change one of these every once in a while actually.
Check the following article and web site for some hints:
Back Pain Mattress Recommendations
Exercise can help a lot to get rid of back problems. Since I've started to work out a few years ago, my chronic pain problems are a thing of the past. I actually just started to work out slowly, not doing anything specific for my back problems, but I do a lot of different things depending on the season, what equipment I have, etc (running, cycling, indoor, outdoor, hiking, weight lifting, rowing, etc). The weight lifting has probably been the most important one that helped me getting rid of the back problems (and it doesn't have to be a bodybuilder style program in case you are concerned about this). Doing only one thing (ie. running) or one type of training (ie. cardio) can actually make things worse. Not to mention that I feel and look a lot better since I started to workout, I am much healthier too. Sorry, this wasn't meant to be a lecture on solving your back problems ;-)