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

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.