Existence and Multiplicity of solutions to some problems in local and nonlocal elliptic PDEs

Abstract

This thesis is about investigating the existence of solutions to some elliptic partial differential
equations and whether they are more than one in number or not. Most of the problems in a nut-shell is like having a local/nonlocal operator with nonlinear terms and a surge term which in most of the cases is a Radon measure. In the thesis we have analyzed the existence of weak
solutions of elliptic partial differential equation which involves either a local operator or a nonlocal operator. We have also proved the multiplicity of solutions (if exists) of elliptic problems and that has been proved by different techniques. Some of the techniques which we have used to prove the existence and multiplicity of solutions are variational technique,Nehari manifold, manifold technique, Banach fixed point theorem, Lion’s concentration compactness principle, etc. Elliptic operators involved in each problem are different and carry different properties.