Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems pp 99-137 | Cite as

# 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.

## Keywords

Elliptic Problem Borel Function Critical Point Theory Hemivariational Inequality Lebesgue Dominate Convergence Theorem## Preview

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