Nonsmooth critical point theory and applications to nonlinear differential problems

The motivation for nonsmooth calculus arises from the limitations of classical anal- ysis, where differentiability assumptions often exclude many relevant models. Real world problems in engineering, economics, and optimization frequently involve ir- regular, discontinuous, or nondifferentiable data. Nonsmooth analysis provides a rigorous framework to extend variational methods and optimization tools beyond the smooth setting. In particular, it establishes a fundamental theoretical basis for the study of variational problems with discontinuous nonlinearities, as well as for prob- lems where the nonlinear term is represented by multifunctions or formulated in the form of hemivariational inequalities.This thesis is situated within this context, devel- oping tools and results concerning the existence and multiplicity of solutions to dif- ferential equations, differential inclusions, and hemivariational inequalities, where classical methods based on differentiability are not applicable. The main tools are Clarke’s generalized gradient and nonsmooth extensions of variational methods, in particular critical point theory. Within this framework, a central role is played by the nonsmooth version of the Mountain Pass Theorem as well as by abstract multiplicity results, such as two and three critical point theorems due to Bonanno, D’Aguì and Winkert [20] and Bonanno and Marano [26], which ensure the existence of multiple solutions under weak regularity and polynomial growth assumptions. The thesis addresses several classes of nonlinear problems. First, the existence of two generalized weak solutions is established in two different settings: an elliptic differential inclusion driven by the Laplace operator with Dirichlet boundary con- ditions, and a second order ordinary differential inclusion with periodic boundary conditions. In both cases, an explicit interval of parameters is identified for which the problem admits a solution. Next, two problems with highly discontinuous nonlinearities are investigated. The first is a Sturm Liouville type problem with sign-changing weight functions and mixed boundary conditions, where the application of a three critical points theo- rem guarantees the existence of at least three generalized solutions. The second one concerns a Neumann problem involving the p-Laplacian and discontinuous nonlin- earities, for which the existence of two weak solutions with opposite energy sign is proven. Finally, a fourth-order hemivariational inequality is studied, and the existence of three distinct solutions is established.