| The goal of this thesis is to present a modification of the Lusternik-Schnirelman category, called f-category, which takes into account a functional f. This f-category will allow us to obtain a louver bound on the number of critical points of f which depends not only on the space as it is the case with the classical category, but also of the functional. This louver bound is always larger or equal and sometimes math bigger than the one obtained in the classical case.; The concepts of relative category, limit relative category and asymptotic category will also be adapted. We will also present the notions of truncated category and of category adapted to a multivalued functional. Thereafter, we will show the relations between the critical points of the function f and the f-category as well as the relation between the f-category and linking sets. Finally, we will present an application to Hamiltonian systems and obtain an existence and multiplicity result.; Keywords. Lusternik-Schnirelman category, relative category, critical points, Hamiltonian systems. |
