Homoclinics for singular strong force Lagrangian systems

Advances in Nonlinear Analysis

article

Marek Izydorek¹  Joanna Janczewska¹  Jean Mawhin² 

[1] Faculty of Applied Physics and Mathematics, Gdańsk University of Technology;Département de mathématique, Université Catholique de Louvain

关键词: homoclinic solution;    homotopy class;    Lagrangian system;    strong force;    rotation number (winding number);   

DOI  :  10.1515/anona-2020-0018

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We study the existence of homoclinic solutions for a class of Lagrangian systems ddt $\begin{array}{} \frac{d}{dt} \end{array} $(∇ Φ ( u̇ ( t ))) + ∇ u V ( t , u ( t )) = 0, where t ∈ ℝ, Φ : ℝ 2 → [0, ∞) is a G -function in the sense of Trudinger, V : ℝ × (ℝ 2 ∖ { ξ }) → ℝ is a C 1 -smooth potential with a single well of infinite depth at a point ξ ∈ ℝ 2 ∖ {0} and a unique strict global maximum 0 at the origin. Under a strong force condition around the singular point ξ , via minimization of an action integral, we will prove the existence of at least two geometrically distinct homoclinic solutions u ± : ℝ → ℝ 2 ∖ { ξ }.

【 授权许可】

CC BY