Abstract

We investigate the singular boundary value problem Δu+u^−γ=0 in D, u=0 on ∂D, where γ>0. For γ>1, we find the estimate |u(x)−b₀δ^2/(γ+1)(x)| <βδ^{(γ−1)/(γ+1)}(x), where b0 depends on γ only, δ(x) denotes the distance from x to ∂D and is β suitable constant. For γ>0, we prove that the function u(1+γ)/2 is concave whenever D is convex. A similar result is well known for the equation Δu+up=0, with 0≤p≤1. For p=0, p=1 and γ≥1 we prove convexity sharpness results.

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