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Homoclinic orbits for second order Hamiltonian equations in $\Bbb R$. (English) Zbl 1264.34090
Author’s abstract: We are concerned with the existence and multiplicity of homoclinic solutions for the second-order Hamiltonian equation $$-\ddot{u}+\omega(t)u=F_u(t,u) \quad t \in \Bbb R, \tag1$$where $\omega \in \mathcal C(\Bbb R)$ is positive and bounded, and $F\in \mathcal C^1(S^1\times\Bbb R)$. Under some growth condition on $F$, we prove that (1) admits at least two solutions which are homoclinic to zero and do not change sign. We also prove that, for every integer $k \geq 1$, (1) possesses at least two solutions homoclinic to zero, changing sign exactly $k$ times, and, for $k \geq 2$, these solutions have at least $k$ and at most $k + 2$ zeros which are isolated, or `isolated from the left’, or `isolated from the right’.

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