Semilinear elliptic inequalities in the exterior of a compact set

TL;DR

This paper investigates conditions under which positive solutions exist for a class of semilinear elliptic inequalities outside a compact set, highlighting the influence of the set's geometry and the functions involved.

Contribution

It provides optimal criteria involving the functions  and f for the existence of smooth positive solutions, considering the geometry of the compact set K.

Findings

Derived necessary and sufficient conditions for solution existence.

Showed the geometry of K significantly affects solutions.

Identified the role of functions  and f in solution behavior.

Abstract

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 $K$.