Abstract
We study a generalized Frenkel-Kontorova model and obtain periodic and heteroclinic mountain pass solutions. The heteroclinic mountain pass solution in the second laminations is new to the generalized Frenkel-Kontorova model. Our proof follows that of Bolotin and Rabinowitz (2005) for an Allen-Cahn equation, which is different from the heat flow method of finding the critical point of the 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.
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Acknowledgements
This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. 34000-31610274). The author wishes to express his gratitude to Professor Zhi-Qiang Wang (Utah State University) for helpful discussion and encouragement. The author also thanks the anonymous referees for their time and patience on carefully reading the manuscript, and pointing out some errors in the manuscript.
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Li, WL. Mountain pass solutions to a generalized Frenkel-Kontorova model. Sci. China Math. 65, 1293–1318 (2022). https://doi.org/10.1007/s11425-020-1794-6
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DOI: https://doi.org/10.1007/s11425-020-1794-6