Journal of Inequalities and Applications
Volume 3 (1999), Issue 4, Pages 313-330
doi:10.1155/S1025583499000223

Qualitative properties of solutions to elliptic singular problems

S. Berhanu,1 F. Gladiali,2 and G. Porru2

1Mathematics Department, Temple University, Philadelphia 19122, PA, USA
2Dipartimento di Matematica, Via Ospedale 72, Cagliari 09124, Italy

Received 23 February 1998; Revised 7 August 1998

Copyright © 1999 S. Berhanu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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)−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.