Berhanu, Shiferaw and Gladiali, Francesca Maria and Porru, Giovanni (1999) Qualitative properties of solutions to elliptic singular problems. Journal of Inequalities and Applications, Vol. 3 (4), p. 313-330. eISSN 1029-242X. Article.
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)−b0δ2/(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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