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There is Bose-Einstein condensate (BEC) for non-interacting boson systems. Can we prove the existence of BEC for interacting systems?

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Bogoliubov proved long, long ago that the condensate is stable against weak interactions. The interactions scatter some fraction of bosons out of the lowest-energy single-particle state ("depleting" the condensate), but off-diagonal long range order remains. For a nice introduction to Bogoliubov's theory see Ben Simon's lectures

http://www.tcm.phy.cam.ac.uk/~bds10/tp3.html

(lecture 9). "Weak" is relative though, and in fact real bose liquids with even decently strong interactions can retain a condensate. Bose condensation turns out to be a very robust phenomenon.

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