A Study on The Existence And Multiplicity Of Solutions To Some Elliptic PDEs Involving Singularity

Abstract

This thesis illuminates on the study of some elliptic partial differential equations (PDEs) involving singularity and measure data or a power non-linearity. The thesis emphasises mostly on the non-local PDEs. The main objective is to obtain the existence, multiplicity and regularity of solutions to the problems considered in the thesis prescribed with certain Dirichlet boundary conditions. Some of the key techniques employed in the thesis to guarantee the existence of solutions are the weak convergence method, Schauder fixed point theorem, Brouwer degree theory, different variants of mountain pass theorem, concentration compactness lemma etc. The existence of infinitely many solutions is accomplished by applying the symmetric mountain pass theorem. The regularity of the solutions is established mostly by the Moser iteration techniques.