A SADDLE POINT THEOREM FOR NON-SMOOTH FUNCTIONALS AND PROBLEMS AT RESONANCE

Abstract. We prove a saddle point theorem for locally Lipschitz functionals with arguments based on a version of the mountain pass theorem for such kind of functionals. This abstract result is applied to solve two different types of multivalued semilinear elliptic boundary value problems with a Laplace–Beltrami operator on a smooth compact Riemannian manifold. The mountain pass theorem of Ambrosetti and Rabinowitz (see [1]) and the saddle point theorem of Rabinowitz (see [18]) are very important tools in the critical point theory of C1-functionals. That is why it is natural to ask what happens if the functional fails to be differentiable. The first who considered such a case were Aubin and Clarke (see [4]) and Chang (see [9]), who gave suitable variants of the mountain pass theorem for locally Lipschitz functionals. For this aim they replaced the usual gradient with a generalized one, which was firstly defined by Clarke (see [10], [11]). In their arguments, the fundamental approach was a “Lipschitz version ” of the deformation lemma in reflexive Banach spaces. In the first part of our paper, after recalling the main properties of the Clarke generalized gradient, we give a variant of the saddle point theorem for locally Lip-schitz functionals. As a compactness condition we use the locally Palais–Smale condition, which was introduced for smooth mappings by Brezis, Coron and Niren-berg (see [7]). We then apply our abstract framework to solve two different types of prob-lems with a Laplace–Beltrami operator on a smooth compact Riemann manifold, possibly with smooth boundary. The first one is related to a multivalued problem with strong resonance at infinity. The literature is very rich in such problems, the first who studied problems at resonance, in the smooth case, being Landesman and Lazer (see [16]). They found sufficient conditions for the existence of solu-tions for some singlevalued equations with Dirichlet conditions. These problems, which arise frequently in mechanics, were thereafter intensively studied and many applications to concrete situations were given. See e.g. [2], [3], [6], [9], [12], [15]