  In this paper, we consider the $\lambda$-model with nearest neighbor
interactions and with competing Potts interactions on the Cayley tree of
order-two. We notice that if $\lambda$-function is taken as a Potts interaction
function, then this model contains as a particular case of Potts model with
competing interactions on Cayley tree. In this paper, we first describe all
ground states of the model. We point out that the Potts model with considered
interactions was investigated only numerically, without rigorous (mathematical)
proofs. One of the main points of this paper is to propose a
measure-theoretical approach for the considered model in more general setting.
Furthermore, we find certain conditions for the existence of Gibbs measures
corresponding to the model, which allowed to establish the existence of the
phase transition.
