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Infinitely many homoclinic orbits for Hamiltonian systems with indefinite sign subquadratic potentials. (English) Zbl 1225.37071

Summary: We deal with the existence and multiplicity of homoclinic solutions of the second-order Hamiltonian system

u ¨(t)-L(t)u(t)+W(t,u(t))=0,

where L(t) and W(t,x) are neither autonomous nor periodic in t. Under the assumption that W(t,x) is indefinite sign and subquadratic as |x|+ and L(t) is a N×N real symmetric positive definite matrices for all t, we establish some existence criteria to guarantee that the above system has at least one or infinitely many homoclinic solutions by using the genus properties in critical theory.

MSC:
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
34C37Homoclinic and heteroclinic solutions of ODE
58E05Abstract critical point theory
70H05Hamilton’s equations
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