A note on homoclinic orbits for second-order Hamiltonian systems
In this note, nonautonomous Hamiltonian systems of the form \[ \ddot{x} - a(t) x + f(t,x) = 0, \quad x \in \mathbb{R}^n, \] are considered. Homoclinic orbits satisfying \(x(t) \to 0\), \(\dot{x}(t) \to 0\), as \(t \to \infty\), are sought for. The type of results obtained are that such homoclinic orbits exist if \(a\) and \(f\) satisfy a number of conditions. In nontechnical terms, these include that \(f\) is sublinear at the origin and asymptotically linear at infinity. Further, \(a(t)\) is large for \(| t| \) large. The results generalize work of \textit{P. H. Rabinowitz} and \textit{K. Tanaka} [Math. Z. 206, No. 3, 473-499 (1991; Zbl 0707.58022)] among others and are proved by variational methods such as the mountain pass theorem.
scientific article; zbMATH DE number 1234933