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

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 $B_1\left(0\right)\backslash \left\{0\right\}$ in $R^N$ ($N\ge 3$) is due to Veron (1981) for $1<q<N∕\left(N-2\right)$, and Brezis-Veron (1980/81) for $q\ge N∕\left(N-2\right)$. In this talk, we extend these results to nonlinear elliptic equations in divergence form $-\nabla \cdot \left(A\left(\left|x\right|\right)\nabla u\right)+u^q=0$ with $q>1$. Here, $A$ denotes a positive $C^1\left(0,1\right]$ function which is regularly varying at zero with index in $\left(2-N,2\right)$. We show that zero is a removable singularity for all positive solutions if and only if $\Phi \notin L^q\left(B_1\left(0\right)\right)$, where $\Phi$ denotes the fundamental solution of $-\nabla \cdot \left(A\left(\left|x\right|\right)\nabla u\right)={\delta }_0$ in the sense of distributions on $B_1\left(0\right)$, and ${\delta }_0$ is the Dirac mass at $0$. We also completely classify the isolated singularities in the more delicate case that $\Phi \in L^q\left(B_1\left(0\right)\right)$. This is joint work with B. Brandolini, F. Chiacchio and C. Trombetti.