Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems pp 67-98 | Cite as

# Variational Methods

## Abstract

This Chapter deals with general techniques for studying the existence and multiplicity of critical points of nondifferentiable functionals in the so-called limit case (see Remark 3.1). There are proved nonsmooth versions of several celebrated results like: Deformation Lemma, Mountain Pass Theorem, Saddle Point Theorem, Generalized Mountain Pass Theorem. First, we present a general deformation result for nonsmooth functionals which can be expressed as a sum of a locally Lipschitz function and a concave, proper, upper semicontinuous function. Then we give a general minimax principle for nonsmooth functionals which can be expressed as a sum of a locally Lipschitz function and a convex, proper, lower semicontinuous functional. Here we are concerned with the limit case (i.e. the equality *c* = *a*, see Remark 3.1), obtaining results which are complementary to the minimax principles in Section 2 of Chapter 2. These general results are applied in the second Section of this Chapter for proving existence, multiplicity and location of solutions to various boundary value and unilateral problems with discontinuous nonlinearities.

## Keywords

Variational Method Critical Point Theory Hemivariational Inequality Finite Covering Mountain Pass Theorem## Preview

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