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

Abstract

In this paper, we are interested in the following damped vibration system: Open image in new window 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(t, x) 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.