Multivalued Elliptic Problems in Variational Form
In Partial Differential Equations, two important tools for proving existence of solutions are the Mountain Pass Theorem of Ambrosetti and Rabinowitz  (and its various generalizations) and the Ljusternik-Schnirelmann Theorem . 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  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.
KeywordsElliptic Problem Borel Function Critical Point Theory Hemivariational Inequality Lebesgue Dominate Convergence Theorem
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