Mountain pass type solutions for a generalized Frenkel-Kontorova model

We study a generalized Frenkel-Kontorova model and obtain periodic and heteroclinic mountain pass solutions. Heteroclinic mountain pass solution in the second laminations is new to the generalized Frenkel-Kontorova model. Our proof follows that of Bolotin and Rabinowitz for an Allen-Cahn equation, which is different with heat flow method for finding critical point of Frenkel-Kontorova model in the literature. The proofs depend on suitable choices of functionals and working spaces. We also study the multiplicity of these mountain pass solutions.