On Singular Solutions of Weighted Divergence Operators

Abstract

We give a complete classification of the isolated singularities of positive solutions to a broad class of nonlinear elliptic equations involving a weighted p-Laplacian and absorption terms in the punctured unit ball centred at zero. We work in the framework of regular variation theory. We introduce the notion of a fundamental solution to our operator, the weighted p-Laplacian. We prove a sharp condition for the removability of all singularities at zero for the positive solutions to our problem. We also show that any non-removable singularity at zero for a positive solution to our prescribed problem is either weak (that is, it is behaves asymptotically like the fundamental solution at zero) or strong (where it dominates the fundamental solution at zero). The main difficulty and novelty of this thesis, for which we develop new techniques, come from the explicit asymptotic behaviour of the strong singularity solutions in the critical case, which had previously remained open even for the p-Laplacian. We also study the existence and uniqueness of the positive solution of our problem with a prescribed admissible behaviour at zero and a Dirichlet condition on the boundary of the unit ball. We also classify the behaviour near zero of the positive solutions with isolated singularities for the weighted p-Laplacian equation. We show that all positive solutions of this problem either has a finite limit at the singularity (and, in certain cases, the solution can be extended as a continuous solution in the entire unit ball), or has a weak singularity depending on the range of p. We note there are no solutions with strong singularities here, unlike the case above where absorption terms are introduced.