Speculating about Mountains

The definition of the weak slope of continuous functions introduced by Degiovanni and Marzocchi (cf. [8]) and its interrelation with the notion “steepness” of locally Lipschitz functions are discussed. A deformation lemma and a mountain pass theorem for usco mappings are proved. The relation between these results and the respective ones for lower semicontinuous functions (cf. [7]) is considered. 0. Introduction. The classical mountain pass theorem of A. Ambrosetti and P.H.Rabinowitz (cf. [1]) has been extended in various directions, one of them being the relaxation of the smoothness assumption on the considered functional. These extensions require notions corresponding to the norm of the derivative (and in particular, to a critical point) in the C1 case. A well known generalization is for locally Lipschitz functionals (cf. [2, 16, 3]) where the most popular notion of the derivative is the Clarke subdifferential (cf. [4]). Recently in [8], [7], [12] and [13] the mountain pass result was extended to the continuous case and a version of this result for lower semicontinuous functionals was proved there. The principal aim of this paper is to propose an alternative approach to the lower semicontinuous case using multivalued usco mappings. 1991 Mathematics Subject Classification: 58E05