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
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
Copyright © 2006 Zhijun Zhang and Jianning Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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,∞)), limx→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.