Elliptic PDEs with Concave and Convex Nonlinearities

Abstract

Elliptic partial differential equations (PDEs) have applications in various fields of science and engineering such as conservation laws, reactiondiffusion problems, thin obstacle problem, phase transitions, crystal dislocation, soft thin films, elastic properties of fractal media and flow through fractal nonsmooth domains. It is also important in various fields of mathematics such as harmonic analysis, differential geometry and calculus of variations. In this thesis, the existence, multiplicity and regularity of solutions to some elliptic PDEs with concave and convex nonlinearities are considered. Existence of solutions are shown for elliptic problems defined on regular and fractal domains subject to different boundary conditions such as Dirichlet, Neumann or nonlinear boundary conditions. Some of the major techniques used in the thesis are: variational methods, weak convergence methods, Mountain pass theorem and its variants, method of subsuper solutions, critical point theory and the theory of monotone operators.