Semilinear elliptic inequalities in the exterior of a compact set

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