The phase transition for the three-state SOS model with one-level competing interactions on the binary tree

In this paper, we consider a three-state solid-on-solid (SOS) model with two competing interactions (nearest-neighbour, one-level next-nearest-neighbour) on the Cayley tree of order two. We show that at some values of parameters the model exhibits a phase transition. We prove that for the model under some conditions there is no antiferromagnetic phases.