On the Boundary Behaviour,Including Second Order Effects,of Solutions to Singular Elliptic Problems期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

On the Boundary Behaviour,Including Second Order Effects,of Solutions to Singular Elliptic Problems

Authors:

Affiliation:

(1) Department of Mathematics, Temple University, Philadelphia, PA 19122, USA;(2) Dipartimento di Matematica e Informatica, University of Cagliari, via Ospedale 72, 09124 Cagliari, Italy

Abstract:

For γ ≥ 1 we consider the solution u = u(x) of the Dirichlet boundary value problem Δu + u ^−γ = 0 in Ω, u = 0 on ∂Ω. For γ = 1 we find the estimate

$u{left( x right)} = p{left( {delta {left( x right)}} right)}{left[ {1 + A{left( x right)}{left( {log frac{1} {{delta {left( x right)}}}} right)}^{{ - in }} } right]},$

where

$$
p{left( r right)} approx r{sqrt {2log {left( {1 mathord{left/
{vphantom {1 r}} right.
kern-nulldelimiterspace} r} right)}} }
$$

near r = 0, δ(x) denotes the distance from x to ∂Ω, 0 < ∈ < 1/2, and A(x) is a bounded function. For 1 < γ < 3 we find

$u{left( x right)} = {left( {frac{{gamma + 1}} {{{sqrt {2{left( {gamma - 1} right)}} }}}delta {left( x right)}} right)}^{{frac{2} {{gamma + 1}}}} {left[ {1 + A{left( x right)}{left( {delta {left( x right)}} right)}^{{2frac{{gamma - 1}} {{gamma + 1}}}} } right]}.$

For γ = 3 we prove that

$u{left( x right)} = {left( {2delta {left( x right)}} right)}^{{frac{1} {2}}} {left[ {1 + A{left( x right)}delta {left( x right)}log frac{1} {{delta {left( x right)}}}} right]}.$

Partially supported by FIRB 2001 and PRIN 2003

Keywords:

elliptic problems singular equations boundary behaviour

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