Variational methods in nonsmooth analysis and quasilinear equations

A natural generalization of the classical theory of critical points is the concept of the theory of critical points for continuous functionals. A fundamental tool in this concept is the weak slope. We introduce it in the first chapter and compare it with other slopes of the literature. We show that this notion is more suitable for the treatment of a certain class of quasilinear equations.Afterwards we formulate some results from regularity theory which we use later for our existence theorems. In Chapter 2 we look for assumptions, under which local minimizers for functionals of the Calculus of Variations in a weak topology are also local minimizers in a strong topology, while in Chapter 3 we give an application of the results of Chapter 2. We prove for a certain class of quasilinear equations the existence of a solution with the help of the method of the sub and super-solutions (Perron). By using a result from Chapter 2 we show that such a solution is a local minimizer for the associated energy-functional. Then we use the Mountain pass principle to conclude the existence of a second solution. The last chapter is concerned with the examinations of the conservation of the critical groups. We use here a generalized Morse Lemma and a Deformation Lemma, which we present and prove