Type "acai berry" into the Google Scholar search engine (scholar.google.com). It will bring up research studies. There are 2, by the same group, and these are the Monavie people. If you look at this site:
http://www.juicescam.com/dr-alex-schauss-orac-and-freeze-dried-acai/ you'll find a discussion as to why the study findings are false.
Acai berries are tasty, and like many other fruits and vegetables they contain high levels of phytochemicals, especially flavonoids. They're also expensive, and they've been subjected to being put on a pedestal by companies in order to make a killing on an exotic new fad.
If you like acai berries, by all means eat them. But don't forget to eat all those other nutritious berries and other fruits and veggies. Call me a cynic, but I've seen so many fads come and go, all in the interest of making money by trying to make us believe that this food is THE miracle food. None of these foods, when consumed as whole foods is bad. In fact, most are great. Are they better than any other highly nutritious plant food? I'd say probably not. It's a diverse, fresh, seasonal, whole foods diet overall that will keep you healthy, not the ingestion of one food.