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

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

Zbl 0788.34042