Semilinear elliptic inequalities in the exterior of a compact set

Abstract

We study the semilinear elliptic inequality $-\Delta u\ge \phi ({\delta }_K(x))f(u)$ in $R^N\setminus K,$ where $\phi ,f$ are non-negative and continuous functions, $K\subset R^N$ $(N\ge 2)$ is a compact set and ${\delta }_K(x)=\mathrm{dist}(x,\partial K)$. We obtain optimal conditions in terms of $\phi$ and $f$ for the existence of $C^2$ positive solutions. Our analysis emphasizes the role played by the geometry of the compact set $K$.

References in corpus (1)

• Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains