The author studies nonautonomous second-order systems of the form
where is a continuous, -periodic real function which may change sign, is a continuous, -periodic positive definite matrix-valued function and satisfies a superquadratic growth condition. Under different sets of additional technical conditions, the author proves the existence of a nontrivial -periodic solution, the existence, for any natural number , of a nontrivial -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
on the spaces , through an application of the mountain pass theorem; the homoclinic solution is found as the limit for of the -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.