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Periodic and homoclinic solutions to a class of Hamiltonian systems with indefinite potential in sign. (English) Zbl 1013.34038

The author studies nonautonomous second-order systems of the form

x ¨-A(t)x+b(t)V ' (x)=0,

where b is a continuous, T-periodic real function which may change sign, A is a continuous, T-periodic positive definite matrix-valued function and VC 2 ( n ,) satisfies a superquadratic growth condition. Under different sets of additional technical conditions, the author proves the existence of a nontrivial T-periodic solution, the existence, for any natural number k, of a nontrivial kT-periodic solution and the existence of one homoclinic solution.

The proofs rely on variational arguments. More specifically, the periodic solutions are found as critical points of the action functionals

f k (u)= 0 kT 1 2 |u ˙| 2 + 1 2 A(t)u,u - b (t) V (u)dt

on the spaces {uH 1 ((0,kT), n ):u(0)=u(kT)}, through an application of the mountain pass theorem; the homoclinic solution is found as the limit for k of the kT-periodic solutions.

Lagrangian systems with a potential changing sign have been considered, e.g., in papers by L. Lassoued [Ann. Mat. Pura Appl., IV. Ser. 156, 73-111 (1990; Zbl 0724.34051)] and M. Girardi and M. Matzeu [NoDEA, Nonlinear Differ. Equa. Appl. 2, No. 1, 35-61 (1995; Zbl 0821.34041)] in connection with the search for periodic solutions, and by P. Caldiroli and P. Montecchiari [Commun. Appl. Nonlinear Anal. 1, No. 2, 97-129 (1994; Zbl 0867.70012)] in connection with the search for homoclinic solutions.

MSC:
34C25Periodic solutions of ODE
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods