Mario Girardi

Some results on solutions of minimal period to superquadratic Hamiltonian systems

Then, it is well known that Rabinowitz’s theorem (see [8]) guarantees the existence, for every T > 0, of at least one nontrivial solution to (l), but he left open the question to find a solution having minimal period T. When H is convex, some results have been recently obtained about such a problem via duality methods. The problem has been completely solved by Ambrosetti & Mancini (see [2,7]), with a suitable hypothesis on the second derivative of H, and by Clarke & Ekeland (see [4]), replacing (Hj), (HA) and (H5) with a subquadratic hypothesis. In another framework, Rabinowitz determines in [9] a family of solutions to (l), having minimal periods which diverge, in the superquadratic case (see theorem 2.34) under the

Dual Morse index estimates for periodic solutions of Hamiltonian systems in some nonconvex superquadratic cases

IN [l], Ekeland and Hofer stated a very general result for the existence of periodic solutions with any prescribed minimal period, for Hamiltonian systems, assuming that the Hamiltonian function H is strictly convex, with superquadratic behaviour at the origin and at infinity. The arguments given in [l] are based on an estimate for the Morse index of a critical point of Mountain Pass type for a functional which is associated with the Hamiltonian system via the duality principle of Clarke and Ekeland [2]. Unfortunately, the functional is only C’ and not C”, then the result is achieved only through a finite dimensional reduction and, as the authors themselves say, “through an inordinate amount of technicalities”. In this paper, we give a “direct” definition of the Morse index of a periodic solution which does not use the duality principle. Indeed, it is based on a study of the Morse indexes of approximating solutions obtained by a Galerkin scheme such as the one introduced by Rabinowitz in [3]. Taking into account some results about linking point theory recently stated in [4-71, we get the same index estimates as in [l], but in a simple way. Moreover, some generalizations are obtained, in the sense that the convexity assumption on H can be replaced either by a local convexity condition or by a condition on the second derivative of H, plus, eventually, a symmetry condition. (See also [8, 91 for some other results obtained in this framework.) In particular, in the locally convex case, the index estimates yield direct information on the minimal period of the corresponding solution. Finally, let us remark that this definition of the Morse index can be adapted to other nonconvex cases, and one can still get some index estimates of the same type.

Essential critical points of linking type and solutions of minimal period to superquadratic Hamiltonian systems

IN A PREVIOUS paper [3] the authors stated a result about the existence of T-periodic solutions (for any T > 0) of autonomous Hamiltonian systems having minimal period T or T/2, in the case where the Hamiltonian function H has a superquadratic behaviour of the type considered by Ekeland and Hofer in [2], but without a global convexity assumption, which is replaced by a local convexity condition. In this paper we use the notion of essentiality for critical points of linking type (see [I, 4, 71 for an exhaustive exposition) in order to construct a T-periodic solution having T as its minimal period. We remark that this result really generalizes the main theorem by Ekeland and Hofer in [2]. The solution is found through a Riesz-Galerkin method as exhibited in [3] (following the basic scheme introduced in [6] by Rabinowitz) and the use of the Marino-Prodi perturbation method. More precisely, as a first step, one finds a critical point of linking type 2, for the restriction F, of F to a suitable finite-dimensional space, for any n E N. Then some suitable estimates for .?,, , derived by the linking construction, enable the use of the Marino-Prodi theorem in order to “approximate” each 2, by a sequence (Zig,“‘], in such a way that .z

SOME RESULTS ON PERIODIC SOLUTIONS OF MOUNTAIN PASS TYPE FOR HAMILTONIAN SYSTEMS

” is a nondegenerate critical point of a functional Fim’, with (Ff”] + F, in the C2-norm, as m + +oo. Indeed 2:“‘) can be chosen as a critical point of linking type satisfying the same estimates as 2, and verifying the further property of essentiality. Roughly speaking, one says that a linking critical point of a functional is not essential or avoidable if, by looking at its linking definition, one can find a local deformation (i.e. in the neighbourhood of w), which enables the construction, for any minimizing sequence of “surfaces” having infinitesimal distances from w, of another minimizing sequence of “surfaces” which keep away from w. The solution z of the Hamiltonian system is found as the limit of a suitable “diagonal” subsequence 1~:: = Zi)j. The fact that the minimal period of z can be either T or T/2 derives from some arguments which were already exploited in [3], based on some estimates for the Morse indexes of finite-dimensional linking points (see [7]). As a final step, one shows that T/2 cannot be a period for z (hence the conclusion that T is the minimal period of z), by showing that the property of T/2-periodicity for z should yield the avoidability of the points of a subsequence of 1~~). First of all, one looks at the maximum negative eigenvalue ,uj of D2F~~‘(zj).

Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign

Some results on the existence of periodic solutions of Hamiltonian systems, having prescribed minimal period, are presented. They are found as critical points of Mountain Pass type of a suitable functional and some estimates on the energy behaviour are shown. The main techniques used are the dual action principle and the Morse index theory.