Mountain pass type solutions for a generalized Frenkel-Kontorova model
Wen-Long Li
TL;DR
This paper investigates mountain pass solutions in a generalized Frenkel-Kontorova model, introducing heteroclinic solutions and employing a novel proof approach inspired by Allen-Cahn equation analysis.
Contribution
It presents the first heteroclinic mountain pass solutions in the second laminations of the generalized Frenkel-Kontorova model, using a new proof method distinct from traditional heat flow techniques.
Findings
Existence of periodic and heteroclinic mountain pass solutions.
Introduction of heteroclinic solutions in the second laminations.
Analysis of the multiplicity of solutions.
Abstract
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.
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Taxonomy
TopicsSolidification and crystal growth phenomena · Advanced Differential Equations and Dynamical Systems · Differential Equations and Numerical Methods