Multiple homoclinic solutions for nonsmooth second-order differential systems (Q6546011)

Homoclinic solutions play an important role in the study of global bifurcations and complex behavior of dynamical systems in different fields of research. But the analysis of the existence of homoclinic orbits is generally a difficult task mathematically.\N\NIn this paper, the author considers the following second-order differential system\N\[\N\ddot{u}(t)+q(t)\dot{u}(t)+\nabla V(t,u(t))=0,\quad t\in \mathbb{R}, \N\] \Nwhere \(q\in \mathcal{C}(\mathbb{R}, \mathbb{R})\), and \(V:\mathbb{R} \times \mathbb{R}^{N}\rightarrow \mathbb{R}\) is a continuous function, differentiable in the second variable with continuous derivative \(\nabla V(t,x)=\frac{\partial V}{\partial x}(t,x)\). When the energy functional associated is not continuously differentiable and does not satisfy the Palais-Smale condition, the author discusses the existence of infinitely many pairs of homoclinic solutions for this nonsmooth second-order differential systems. Some known results are extended.