Infinitely many homoclinic solutions for a class of damped vibration problems

• Published in: Electronic Journal of Qualitative Theory of Differential Equations
• Main Authors: Huijuan Xu, Shan Jiang, Guanggang Liu
• Format: Article
• Language: English
• Published: University of Szeged 2022-12-01
• Subjects: homoclinic solutions; clark's theorem; critical points; palais–smale condition
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=9989
• ISSN: 1417-3875

In this paper, we consider the multiplicity of homoclinic solutions for the following damped vibration problems $$ \ddot{x}(t)+B\dot{x}(t)-A(t)x(t)+H_{x}(t,x(t))=0,$$ where $A(t)\in (\mathbb{R},\mathbb{R}^{N})$ is a symmetric matrix for all $t\in \mathbb{R}$, $B=[b_{ij}]$ is an antisymmetric $N \times N$ constant matrix, and $H(t,x)\in C^{1}(\mathbb{R}\times B_{\delta},\mathbb{R})$ is only locally defined near the origin in $x$ for some $\delta>0$. With the nonlinearity $H(t,x)$ being partially sub-quadratic at zero, we obtain infinitely many homoclinic solutions near the origin by using a Clark's theorem.