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

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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$” 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).