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 - <p style="text-align: justify;">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 $b_2 ≥b_1 &gt;0$ 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.</p>