Multitransition solutions for a generalized Frenkel-Kontorova model

Wen-Long Li, ,School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China, Xiaojun Cui, ,Department of Mathematics, Nanjing University, Nanjing 210093, China
2020 Discrete and Continuous Dynamical Systems. Series A  
We study a generalized Frenkel-Kontorova model. Using minimal and Birkhoff solutions as building blocks, we construct a lot of homoclinic solutions and heteroclinic solutions for this generalized Frenkel-Kontorova model under gap conditions. These new solutions are not minimal and Birkhoff any more. We use constrained minimization method to prove our results. 2020 Mathematics Subject Classification. 70G75, 74G22, 74G35. By Lemma 2.2, Adding the above two inequalities and letting q → ∞, we get
more » ... us letting k → ∞ shows that Finally, letting p → ∞ and then → 0 yields This proves (47) and completes the proof of d 1 (v 0 , w 0 ) > c 1 (v 0 , w 0 ). The other inequality d 1 (w 0 , v 0 ) > c 1 (w 0 , v 0 ) can be proved similarly. The following result gives that minimal solutions of (6) inΓ(v 0 , w 0 ) are Birkhoff. It will be used to show that our solution obtained in Theorem 2.9 is not minimal any more. Proposition 5. If u ∈Γ 1 (v 0 , w 0 ) is minimal, u is Birkhoff.
doi:10.3934/dcds.2020273 fatcat:gklmj3unkjbu3iuwwwzp4reyui