Multivalued Elliptic Problems in Variational Form

Abstract

In Partial Differential Equations, two important tools for proving existence of solutions are the Mountain Pass Theorem of Ambrosetti and Rabinowitz [1] (and its various generalizations) and the Ljusternik-Schnirelmann Theorem [16]. These results apply to the case when the solutions of the given problem are critical points of an appropriate functional of energy f, which is supposed to be real and C ¹, or only differentiable, on a real Banach space X. One may ask what happens if f, which often is associated to the original equation in a canonical way, fails to be differentiable. In this case the gradient of f must be replaced by a generalized one, which is often that introduced by Clarke in the framework of locally Lipschitz functionals. In this setting, Chang [4] was the first who proved a version of the Mountain Pass Theorem, in the case when X is reflexive. For this aim, he used a “Lipschitz version” of the Deformation Lemma. The same result was used for the proof of the Ljusternik-Schnirelmann Theorem in the locally Lipschitz case. As observed by Brézis, the reflexivity assumption on X is not necessary.