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

Zhijun Zhang1 and Jianning Yu2

 1Department of Mathematics and Informational Science, Yantai University, Yantai, Shandong 264005, China
2College of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China
Received 23 September 2005; Revised 10 November 2005; Accepted 15 November 2005

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.