A Study on Singular and Non-singular Elliptic and Parabolic PDEs

Abstract

The thesis is concerned about singular elliptic, parabolic partial differential equations (PDEs) and non-singular elliptic PDEs. A problem having nonlinearities that blow up to infinity near zero is considered as a ‘singular problem’. Most of the problems in the thesis are involved with nonlocal elliptic operators such as the fractional order Laplacian or p-Laplacian with constant or variable exponents. The objective of the thesis is to investigate the existence and multiplicity/unicity of weak solutions to PDEs involving a singular nonlinearity or/and critical nonlinearities or/and an L1/measure datum, assigned with Dirichlet boundary conditions. The approach varies from problem to problem. Variational techniques comprising of the variants of mountain pass theorem, the Ekeland’s variational principle are the commonly used techniques in the thesis. The PDEs involving a singular nonlinearity, a Radon measure or an L1 function are approached by the weak convergence method. The concentration compactness principles and the method of layer potentials are used to tackle the problems with critical exponents and the problems with mixed boundary value type respectively. A concentration compactness type principle in two critical exponent set up is also derived. The solutions are found to be in fractional or integer order Sobolev spaces with constant or variable exponents. Furthermore, various notions of solutions like entropy solution, very weak solution, etc. are also introduced for a few singular problems. The presence of irregular data like L1/measure data affects the regularity of solutions, i.e. the solutions are obtained as limits of approximations (SOLA) with a lesser degree of differentiability or/and integrability.