  In the present paper we study a phase transition problem for the Potts model
with three competing interactions, the nearest neighbors, the second neighbors
and triples of neighbors and non-zero external field on Cayley tree of order
two. We prove that for some parameter values of the model there is phase
transition. We reduce the problem of describing by limiting Gibbs measures to
the problem of solving a system of nonlinear functional equations. We extend
the results obtained by Ganikhodjaev and Rozikov [Math. Phys. Anal. Geom.,
2009, 12, No. 2, 141-156] on phase transition for the Ising model to the Potts
model setting.
