Boundary Value Problems
Volume 2006 (2006), Article ID 75674, 10 pages

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

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.


By Karamata regular variation theory, we show the existence and exact asymptotic behaviour of the unique classical solution uC2+α(Ω)C(Ω¯) near the boundary to a singular Dirichlet problem Δu=g(u)k(x), u>0, xΩ, u|Ω=0, where Ω is a bounded domain with smooth boundary in N, gC1((0,),(0,)), limx0+(g(ξt)/g(t))=ξγ, for each ξ>0 and some γ>1; and kClocα(Ω) for some α(0,1), which is nonnegative on Ω and may be unbounded or singular on the boundary.