Semilinear elliptic inequalities in the exterior of a compact set

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We study the semilinear elliptic inequality -Δu≥φ(δₖ(x))f(u) in Rⁿ K, where φ, f are non-negative and continuous functions, K⊂ Rⁿ (N≥ 2) is a compact set and δₖ(x)= dist(x,∂ K). We obtain optimal conditions in terms of φ and f for the existence of C² positive solutions. Our analysis emphasizes the role played by the geometry of the compact set K.