|Title||On singular solutions of weighted divergence operators|
|Presenter||Ting-Ying Chang (The University of Sydney)|
|Session||Analysis and Partial Differential Equations|
We study the behaviour of solutions with isolated singularities to weighted $p$-Laplacian operators in the punctured unit ball centred at zero. We impose our weight to be in the framework of regular variation. We are able to classify the solutions to our for the entire range of $p$ by adapting the rescaling method from Kichenassamy and V\'eronís study of isolated singularities of $p$-harmonic functions. The adapted method relies on the construction of a suitable "fundamental solution" depending on the range of $p$. We show that all possible singularities at $0$ for a positive solution of our problem are either removable (and the solution can be extended as a continuous solution in the entire ball), weak, or the solution can be extended as a continuous function in the entire ball. We note there exists no solutions with strong singularities to our problem, a case which only arises when absorption terms are present.
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