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Zbl 0759.58018
Omana, W.; Willem, M.
Homoclinic orbits for a class of Hamiltonian systems. (English)
[J] Differ. Integral Equ. 5, No.5, 1115-1120 (1992). ISSN 0893-4983

The Hamiltonian system under consideration is governed by equations of the form $$\ddot q+V\sb q(t,q)=\ddot q-L(t)q+W\sb q(t,q)=0,$$ where $L(t)$ is a positive definite matrix and further technical conditions, among other things, ensure that the origin is a local maximum of $V$ for all $t$. The authors first reconsider a theorem by Rabinowitz and Tanaka concerning the existence of a homoclinic orbit emanating from 0. Using a new compact imbedding theorem, they are able to show that the Palais- Smale condition is satisfied, which in turn makes it possible to prove the above cited theorem by the more traditional techniques relying on the Mountain Pass Theorem. If, in addition, $W$ is an even function for all $t$, they make use of the symmetric mountain pass theorem to prove the existence of an unbounded sequence of homoclinic orbits.
[W.Sarlet (Gent)]

MSC 2000:: *37J99 Finite-dimensional Hamiltonian etc. systems
34C37 Homoclinic and heteroclinic solutions of ODE
58E05 Abstract critical point theory

Keywords: Hamiltonian systems; mountain pass theorem; homoclinic orbits

Cited in: Zbl 1073.34046 Zbl 0759.58016

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