To put things in perspective here, I bought a new Raleigh Sojourn a while back. It came with Vittoria Randonneur Cross tires in 35x700 size. I now have 12,900 miles on that bike, am on my 3rd rear tire and 2nd front tire, and have zero flats in that time. So if you're trying to eliminate flats, consider that there may be better alternatives out there.
I have read widely varying reports on "airless" tires and the companies that sell them, and taken as a whole, have never had any desire to persue the subject further.
One thing to note is that using a solid tire changes the stress distribution in the rim, so you may have wheel problems instead of tire problems.