**SMS scnews item created by Daniel Daners at Tue 11 Sep 2012 0847**

Type: Seminar

Modified: Tue 11 Sep 2012 0902; Tue 11 Sep 2012 0905

Distribution: World

Expiry: 17 Sep 2012

**Calendar1: 17 Sep 2012 1400-1500**

**CalLoc1: AGR Carslaw 829**

Auth: daners@bari.maths.usyd.edu.au

### PDE Seminar

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

### Cirstea

Florica Cîrstea

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 \(-\Delta u+u^q=0\) in the punctured unit
ball \(B_1(0)\setminus \{0\}\) in \(R^N\) (\(N\geq 3\)) 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 \(\Phi\not\in L^q(B_1(0))\),
where \(\Phi\) denotes the fundamental solution of
\(-\nabla\cdot(A(|x|)\nabla u)=\delta_0\) 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 \(\Phi\in L^q(B_1(0))\). This is joint work
with B. Brandolini, F. Chiacchio and C. Trombetti.

Check also the PDE
Seminar page. Enquiries to Florica
Cîrstea or Daniel Daners.