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.
Qualitative properties of solutions to elliptic singular problems / Berhanu, Shiferaw; Porru, Giovanni; Gladiali, Francesca Maria. - 3:4(1999), pp. 313-330. [10.1155/S1025583499000223]
Qualitative properties of solutions to elliptic singular problems
Gladiali, Francesca Maria
1999-01-01
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.File | Dimensione | Formato | |
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