A Birkhoff-Lewis-type theorem for some Hamiltonian PDEs

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Silvia Cingolani. Monica Lazzo. When the problem has a suitable symmetry, namely V is even, a Z2 -version of Ljusternik—Schnirelman theory can be applied. Key words and phrases.

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Periodic solutions, singular potentials, winding number. Supported by M. Cingolani — M. Lazzo the plane, it has a finite Z2 -index; as far as large period solutions are concerned, this allows avoiding trivial solutions. We refer to Remark 5. In the general case, P has still an intrinsic symmetry, i.

In order to prove existence of multiple nontrivial periodic solutions of large period to P , without evenness assumptions on V , our idea consists in regarding P as a problem with a singular potential.

Let us recall that existence and multiplicity of periodic solutions with singular potential have been extensively investigated by many authors; for instance, see [1], [4], [9] for strong-force potential, [7], [12], [13] for weak-force potential and references therein. Before stating our main results, we need some notations and definitions. Remark that, by its very definition, an admissible set is not simply connected in the plane. Theorem 1. We point out that in Theorem 1. Such a geometrical hypothesis has two main consequences. Firstly, it guarantees the existence of a comparison term, in the sense of the previous remark.

Secondly, it allows an easy modification of V in U such that the solutions of the modified equation still solve PT. Let us remark that in Theorem 1.

In Theorem 1. Finally, in this paper we describe the qualitative behaviour of T -periodic solutions to P as the period becomes larger and larger. Precisely, we show that if xT is any solution found via Theorem 1. Variational setting and preliminaries Let us introduce some notations. We aim to find critical points of fT as minima in suitable homotopy classes and to this end we shall adapt the arguments in [6].

Section 1 for notations ; therefore, we can consider the winding number of x around any such point, which we shall denote by Ind x. Lemma 2. The Kourovka Notebook , arXiv : Barwise , S. Feferman , eds.

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Journal of Symbolic Logic Volume. Fundamenta Mathematicae. Bibcode : math Classification theory for abstract elementary classes. College Publications. July 24, Categoricity PDF.

Introduction

American Mathematical Society. Archived PDF from the original on July 29, Retrieved February 20, Bibcode : arXiv Journal of Symbolic Logic. May Journal of Combinatorial Theory, Series B. Foreman, Banff, Alberta, Die Welt der Primzahlen.


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