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Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities. (English) Zbl 1218.37082
The authors prove the existence of infinitely many homoclinic (to 0) solutions for the second order system $\ddot u-L(t)u+W_u(t,u) = 0$ under the conditions that $L=L(t)$ is a symmetric matrix such that the smallest eigenvalue of $L(t)$ tends to infinity as $|t|\to\infty$, and the function $W$ is even in $u$ and either sub- or superquadratic as $u\to 0$ and $|u|\to\infty$. In the subquadratic case, an Ahmad-Lazer-Paul type condition is imposed and for superquadratic $W$ a condition which is weaker than that of Ambrosetti and Rabinowitz is assumed. The proofs are carried out by verifying the assumptions of two versions of the fountain theorem, respectively due to Bartsch and Willem, and to Zou. Necessary compactness of the Euler-Lagrange functional corresponding to the problem is a consequence of the fact that the smallest eigenvalue of $L(t)$ goes to infinity with $|t|$.

37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
34C37Homoclinic and heteroclinic solutions of ODE
58E05Abstract critical point theory
Full Text: DOI
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