Boundary Value Problems
Volume 2006 (2006), Article ID 75674, 10 pages
doi:10.1155/BVP/2006/75674

The exact asymptotic behaviour of the unique solution to a singular Dirichlet problem

Abstract

By Karamata regular variation theory, we show the existence and exact asymptotic behaviour of the unique classical solution u∈C2+α(Ω)∩C(Ω¯) near the boundary to a singular Dirichlet problem −Δu=g(u)−k(x), u>0, x∈Ω, u|∂Ω=0, where Ω is a bounded domain with smooth boundary in ℝN, g∈C1((0,∞),(0,∞)), lim⁡x→0+(g(ξt)/g(t))=ξ−γ, for each ξ>0 and some γ>1; and k∈Clocα(Ω) for some α∈(0,1), which is nonnegative on Ω and may be unbounded or singular on the boundary.