Dead cores and bursts for quasilinear singular elliptic equations

We consider divergence structure quasilinear singular elliptic partial differential equations on domains of R-n and show that there exist solutions with dead cores and, furthermore, solutions which involve both a dead core and bursts within the core. The results are obtained under appropriate monotonicity conditions on both the nonlinearity and the elliptic operator. Important special cases treated here are the p-Laplace and the mean curvature operators. We also study related problems for p-Laplace equations with weights, which include the Matukuma equation as a prototype. While it is usually thought that dead cores arise due to loss of smoothness of the underlying equation, we show by examples that they can occur equally for analytic p-Laplace equations.