In this research, we consider the λ-model with and without competing interaction on Cayley tree of order two. Description of ground states becomes one of the main elements to study as phase diagram of Gibbs measure for a Hamiltonian is close to the phase diagram of isolated ground states of the Hamiltonian. For the λ-model on infinite Cayley tree, we describe the set of periodic and weakly periodic ground states corresponding normal subgroup of the Cayley tree group representation. We construct 81 different combination of configurations and classify the configurations under 10 different regions so that the configurations will achieve ground states. We describe periodic and weakly periodic ground states for the considered model by using periodic and weakly periodic configurations. For the second result of the research, we consider λ-model with competing Potts interaction on Cayley tree of order two. As explained in previous section, we describe the periodic ground states for the considered model. Note that for this model, we have 12 different regions for the configurations to achieve ground states. For some domain of interactions strength, the configuration of periodic ground states cannot be achieved. By using Kolmogorov criteria, Gibbs measures for this model was described by deriving infinite volume distribution using given finite-dimensional distributions and find the probability measures with given conditional probability. By considering translation invariant Gibbs measure, we analyse the system of equations derived and study the phase transition phenomenon by proving the existence of multiple translation-invariant solutions for the system of equations. Phase transitions occurs if there exist two or more solutions.