On the Linking Principle

Abstract

We give a linking theorem that strengthens and unifies some many minimax theorems including Ambrosetti-Rabinowitz ``mountain pass theorem'', Rabinowitz ``multidimensional mountain pass theorem'', Rabinowitz ``saddle point theorem'' and Silva's variants of these results. We focus our attention especially on ``the limiting case'', known to be true for the mountain pass principle, where some information on the location of the critical points is given. Two forms of this theorem are given: the first one is established via a deformation lemma and we use Ekeland's variational principle to get the second one.