On homoclinic and heteroclinic orbits of Hamiltonian systems. (English) Zbl 0897.34045

Korman, P.; Lazer, A. C.; Li, Y.

Differ. Integral Equ. 10, No. 2, 357-368 (1997).

The authors give sufficient conditions for the existence of homoclinic and heteroclinic orbits of Hamiltonian systems \[ u''-L(t)u+V_u(t,u)=0,\tag{*} \] where \(L\) is a symmetric positive definite \(n\times n\) matrix and the potential \(V\) is supposed to be superquadratic in \(u\). The system (*) is first studied on a bounded interval \((-T,T)\) with the boundary conditions \(u(-T)=0=u(T)\) (the existence of a nontrivial solution is proved via the mountain pass lemma) and then the limiting process \(T\to \infty \) is used. A similar approach to an investigation of orbits of (*) is used by P. Korman and A. C. Lazer [Electron. J. Differ. Equ., 1994/01 (1994; Zbl 0788.34042)] but under the assumption that \(L\) and \(V\) are even functions of \(t\).

Reviewer: O.Došlý (Brno)

MSC:

34C37

Homoclinic and heteroclinic solutions to ordinary differential equations

Keywords:

homoclinic orbit; heteroclinic orbit; mountain pass lemma; Hamiltonian system