Second order estimates for large solutions of elliptic equations

Abstract

This paper deals with the second term asymptotic behavior of large solutions to the problems Δ u = b ( x ) f ( u ) , x ∈ Ω , subject to the singular boundary condition u ( x ) = ∞ , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R N , and b ( x ) is a non-negative weight function. The absorption term f is regularly varying at infinite with index ρ > 1 (that is lim u → ∞ f ( ξ u ) / f ( u ) = ξ ρ for every ξ > 0 ) and the mapping f ( u ) / u is increasing on ( 0 , + ∞ ) . Our analysis relies on the Karamata regular variation theory.