Local behaviour of singular solutions for nonlinear elliptic equations in divergence form

Florica C�rstea
The University of Sydney
17 Sep 2012, 2-3pm, Carslaw Room 829 (AGR)

Abstract

A complete classification of the behaviour near zero of all non-negative solutions of $-\mathrm{\Delta }u+u^q=0$ in the punctured unit ball $B_1(0)\setminus \{0\}$ in $R^N$ ($N\ge 3$) is due to Veron (1981) for \(11\). Here, $A$ denotes a positive $C^1(0,1]$ function which is regularly varying at zero with index in $(2-N,2)$. We show that zero is a removable singularity for all positive solutions if and only if $\mathrm{\Phi }\notin L^q(B_1(0))$, where $\mathrm{\Phi }$ denotes the fundamental solution of $-\mathrm{\nabla }\cdot (A(|x|)\mathrm{\nabla }u)={\delta }_0$ in the sense of distributions on $B_1(0)$, and ${\delta }_0$ is the Dirac mass at $0$. We also completely classify the isolated singularities in the more delicate case that $\mathrm{\Phi }\in L^q(B_1(0))$. This is joint work with B. Brandolini, F. Chiacchio and C. Trombetti.

Check also the PDE Seminar page. Enquiries to Florica C�rstea or Daniel Daners.