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On Ground-State Homoclinic Orbits of a Class of Superquadratic Damped Vibration Systems

  • Mohsen Timoumi
Article
  • 41 Downloads

Abstract

In this paper, we are interested in the following damped vibration system: where B is an antisymmetric \(N\times N\) constant matrix, \(q:{\mathbb {R}}\longrightarrow {\mathbb {R}}\) is a continuous function, \(L(t)\in C({\mathbb {R}},{\mathbb {R}}^{N^{2}})\) is a symmetric matrix, and \(W(t,x)\in C^{1}({\mathbb {R}}\times {\mathbb {R}}^{N},{\mathbb {R}})\) are neither autonomous nor periodic in t. The novelty of this paper is that, supposing that \(Q(t)=\int ^{t}_{0}q(s)\mathrm{d}s\) is bounded from below and L(t) is coercive unnecessarily uniformly positively definite for all \(t\in {\mathbb {R}}\), we establish the existence of ground-state homoclinic solutions for (1) when the potential W(tx) satisfies a kind of superquadratic conditions due to Ding and Luan for Schr\({\ddot{o}}\)dinger equation. The main idea here lies in an application of a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou. Some recent results in the literature are generalized and significantly improved.

Keywords

Damped vibration systems ground-state homoclinic solutions superquadradicity variational methods weak linking theorem 

Mathematics Subject Classification

37J45 

Notes

Acknowledgements

The author thanks the referee for valuable comments and suggestions that improved the paper.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Monastir Faculty of SciencesMonastirTunisia

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