This volume collects two articles by Christine Laurent-Thiébaut and Jürgen Leiterer which were submitted to, and accepted by the Annali della Scuola Normale Superiore, Classe di Scienze: The q-convex case; the q-concave case. Owing to the character of the systematic exposition of the new scientific results achieved and to the size of the work, the authors agreed to have the two papers published in a separate volume.