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The existence and asymptotic behaviour of the unique solution near the boundary to a singular Dirichlet problem with a convection term

Published online by Cambridge University Press:  12 July 2007

Zhijun Zhang
Affiliation:
Department of Mathematics and Informational Science, Yantai University, Yantai, Shandong 264005, People's Republic of China (zhangzj@ytu.edu.cn)
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Abstract

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We show the existence and exact asymptotic behaviour of the unique solution uC2(Ω)∩C(Ω̄) near the boundary to the singular nonlinear Dirichlet problem −Δu = k(x)g(u) + λ|∇u|q, u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded domain with smooth boundary in RN, λ ∈ R, q ∈ [0, 2], g(s) is non-increasing and positive in (0, ∞), lims→0+g(s) = +∞, kCα(Ω) is non-negative non-trivial on Ω, which may be singular on the boundary.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2006