Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials
Section snippets
Introduction and main results
In this paper, we consider the second-order Hamiltonian system: where is a symmetric matrix valued function, , and . We say that a solution of (1.1) is homoclinic (to 0) if and as . With the aid of the variational methods, many authors extensively investigated the existence of (multiple) homoclinic solutions for problem (1.1), see [2], [3], [4], [5], [6]. Most of them treated the
Proof of Theorem 1
We denote by the self-adjoint extension of the operator with the domain . Let , the domain of , and define on the inner product and norm , where denotes the inner product of . Then is a Hilbert space.
It is easy to prove that the spectrum consists of eigenvalues numbered by (counted in their multiplicities) and a corresponding system of eigenfunctions which forms
Acknowlegements
The authors of this paper would like to express their sincere thanks to the referee for useful suggestions. Especially, they would also like to thank Guihua Fei and Ou-Tang whose papers drew their attention to this problem.
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