Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. (English) Zbl 0744.34045

Coti Zelati, Vittorio; Rabinowitz, Paul H.

J. Am. Math. Soc. 4, No. 4, 693-727 (1991).

The goal of this paper is to establish the existence of infinitely many homoclinic orbits for a class of second order Hamiltonian systems of the form (HS) \(\ddot q-L(t)q+V_ q(t,q)=0\) under suitable assumptions, in particular \(V\in C^ 2\) (\(\mathbb{R}\times\mathbb{R}^ n,\mathbb{R}\)) is of kind “superquadratic”. The approach to (HS) involves the use of variational methods of a minimax nature and the authors seek solutions of (HS) as critical points of the functional \(I\) defined on \(W^{1,2}(\mathbb{R};\mathbb{R}^ n)\) by \(I(q)=\int^ \infty_{-\infty}({1\over 2}| \dot q(t)|^ 2+L(q)q.q-V(t,q))dt\).

Reviewer: P.Bolley (Nantes)

MSC:

34C37	Homoclinic and heteroclinic solutions to ordinary differential equations
58E99	Variational problems in infinite-dimensional spaces
37-XX	Dynamical systems and ergodic theory

Keywords:

homoclinic orbits; second order Hamiltonian systems; variational methods; minimax