Homoclinics for second order conservative systems. (English) Zbl 0804.34046

Ambrosetti, A.; Bertotti, M. L.

Miranda, Mario (ed.), Partial differential equations and related subjects. Proceedings of the conference dedicated to Louis Nirenberg held in Trento, Italy, September 3-8, 1990. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 269, 21-37 (1992).

The authors consider a conservative system \(q''+ V_ q(q)=0\), \(q\in \mathbb{R}^ n\). It is assumed that the origin is an equilibrium. Usually to prove existence of homoclinic orbits of 0 “superquadraticity” or convexity of the nonlinear part of the vectorfield is assumed. The authors obtain an existence result for homoclinics dropping such a global condition. They also consider singular potentials.
For the entire collection see [Zbl 0785.00033].

Reviewer: S.Yu.Pilyugin (St.Peterburg)

MSC:

34C37

Homoclinic and heteroclinic solutions to ordinary differential equations

Keywords:

conservative system; homoclinic orbits; existence; singular potentials