Dead Cores and Bursts for Quasilinear Singular Elliptic Equations

We consider divergence structure quasilinear singular elliptic partial differential equations on domains of Rn 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.