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

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

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

For γ = 3 we prove that

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