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Volume 41, Issue 2
Homoclinic Solutions for a Class of Hamiltonian Systems with Small External Perturbations

Wenzhuang Zhu, Chunhui Hu & Shuguan Ji

DOI: 10.4208/cmr.2025-0016

Commun. Math. Res., 41 (2025), pp. 148-172.

Published online: 2025-06

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  • Abstract

This paper is concerned with the existence of nontrivial homoclinic solutions for a class of second order Hamiltonian systems with external forcing perturbations $\ddot{q}+A\dot{q}+V_q(t,q)= f(t),$ where $q= (q_1,q_2,···,q_N)∈\mathbb{R}^N,$ $A$ is an antisymmetric constant $N×N$ matrix, $V(t,q) = −K(t,q)+W(t,q)$ with $K,W ∈ C^1 (\mathbb{R},\mathbb{R}^N)$ and satisfying $b_1|q|^2 ≤ K(t,q) ≤ b_2|q|^2$ for some positive constants $b_2 ≥b_1 >0$ and external forcing term $f ∈C(\mathbb{R},\mathbb{R}^N)$ being small enough. Under some new weak superquadratic conditions for $W,$ by using the mountain pass theorem, we obtain the existence of at least one nontrivial homoclinic solution.

  • Keywords

Homoclinic solution, Hamiltonian system, mountain pass theorem, critical point.

  • AMS Subject Headings

58E05, 58E50, 34C37

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-41-148, author = {Zhu , WenzhuangHu , Chunhui and Ji , Shuguan}, title = {Homoclinic Solutions for a Class of Hamiltonian Systems with Small External Perturbations}, journal = {Communications in Mathematical Research }, year = {2025}, volume = {41}, number = {2}, pages = {148--172}, abstract = {

This paper is concerned with the existence of nontrivial homoclinic solutions for a class of second order Hamiltonian systems with external forcing perturbations $\ddot{q}+A\dot{q}+V_q(t,q)= f(t),$ where $q= (q_1,q_2,···,q_N)∈\mathbb{R}^N,$ $A$ is an antisymmetric constant $N×N$ matrix, $V(t,q) = −K(t,q)+W(t,q)$ with $K,W ∈ C^1 (\mathbb{R},\mathbb{R}^N)$ and satisfying $b_1|q|^2 ≤ K(t,q) ≤ b_2|q|^2$ for some positive constants $b_2 ≥b_1 >0$ and external forcing term $f ∈C(\mathbb{R},\mathbb{R}^N)$ being small enough. Under some new weak superquadratic conditions for $W,$ by using the mountain pass theorem, we obtain the existence of at least one nontrivial homoclinic solution.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2025-0016}, url = {http://global-sci.org/intro/article_detail/cmr/24189.html} }
TY - JOUR T1 - Homoclinic Solutions for a Class of Hamiltonian Systems with Small External Perturbations AU - Zhu , Wenzhuang AU - Hu , Chunhui AU - Ji , Shuguan JO - Communications in Mathematical Research VL - 2 SP - 148 EP - 172 PY - 2025 DA - 2025/06 SN - 41 DO - http://doi.org/10.4208/cmr.2025-0016 UR - https://global-sci.org/intro/article_detail/cmr/24189.html KW - Homoclinic solution, Hamiltonian system, mountain pass theorem, critical point. AB -

This paper is concerned with the existence of nontrivial homoclinic solutions for a class of second order Hamiltonian systems with external forcing perturbations $\ddot{q}+A\dot{q}+V_q(t,q)= f(t),$ where $q= (q_1,q_2,···,q_N)∈\mathbb{R}^N,$ $A$ is an antisymmetric constant $N×N$ matrix, $V(t,q) = −K(t,q)+W(t,q)$ with $K,W ∈ C^1 (\mathbb{R},\mathbb{R}^N)$ and satisfying $b_1|q|^2 ≤ K(t,q) ≤ b_2|q|^2$ for some positive constants $b_2 ≥b_1 >0$ and external forcing term $f ∈C(\mathbb{R},\mathbb{R}^N)$ being small enough. Under some new weak superquadratic conditions for $W,$ by using the mountain pass theorem, we obtain the existence of at least one nontrivial homoclinic solution.

Zhu , WenzhuangHu , Chunhui and Ji , Shuguan. (2025). Homoclinic Solutions for a Class of Hamiltonian Systems with Small External Perturbations. Communications in Mathematical Research . 41 (2). 148-172. doi:10.4208/cmr.2025-0016
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