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Homoclinic orbits for second order Hamiltonian equations in . (English) Zbl 1264.34090

Author’s abstract: We are concerned with the existence and multiplicity of homoclinic solutions for the second-order Hamiltonian equation

-u ¨+ω(t)u=F u (t,u)t,(1)

where ω𝒞() is positive and bounded, and F𝒞 1 (S 1 ×). 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 k1, (1) possesses at least two solutions homoclinic to zero, changing sign exactly k times, and, for k2, 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’.

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