On the Linking Principle Youssef Jabri and Mimoun Moussaoui

During the last twenty years, many minimax theorems that have proved to be very useful tools in finding critical points of functionals have been established. They have all in common a geometric intersection property known as the linking principle. Our purpose in this paper is to give a linking theorem that strengthens and unifies some of these works. We think essentially to Ambrosetti-Rabinowitz “mountain pass theorem” [1], Rabinowitz “multidimensional mountain pass theorem” [2], Rabinowitz “saddle point theorem” [3] and Silva’s variants of these results [4]. We focus our attention especially on “the limiting case”, known to be true for the mountain pass principle [5], where some information on the location of the critical points is given. We give two forms of this theorem, in the first part of the paper, the first one is established via a deformation lemma and in the second part we use Ekeland’s variational principle to get the second one. Unfortunately, we could not remove the finite dimension condition that appears in the results [5] and [6]. This finite dimension assumption is dropped only when dealing with a special kind of functionals having a predefined shape [7, 8, 4].