The exact asymptotic behaviour of the unique solution to a singular Dirichlet problem
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By Karamata regular variation theory, we show the existence and exact asymptotic behaviour of the unique classical solution u ∈ C 2+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 ∈ C 1 ((0, ∞),(0, ∞)), limt→0+ (g(ξt)/g(t)) = ξ −γ , α (Ω) for some α ∈ (0,1), which is nonnegative for each ξ > 0 and some γ > 1; and k ∈ Cloc on Ω and may be unbounded or singular on the boundary. Copyright © 2006 Z. Zhang and J. 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. 1. Introduction and the main results The purpose of this paper is to investigate the existence and exact asymptotic behaviour of the unique classical solution near the boundary to the following model problem: −u = g(u) − k(x),
u > 0, x ∈ Ω, u|∂Ω = 0,
(1.1)
α (Ω) for where Ω is a bounded domain with smooth boundary in RN (N ≥ 1), k ∈ Cloc some α ∈ (0,1), which is nonnegative on Ω, and g satisfies (g1 ) g ∈ C 1 ((0, ∞),(0, ∞)), g (s) ≤ 0 for all s > 0, lims→0+ g(s) = +∞. The problem arises in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrical conductive materials (see [4, 7, 12, 14]). The main feature of this paper is the presence of the two terms, the singular term g(u) which is regular varying at zero of index −γ with γ > 1 and includes a large class of singular functions, and the nonhomogeneous term k(x), which may be singular on the boundary. This type of nonlinear terms arises in the papers of D´ıaz and Letelier [6], Lasry and Lions [10] for boundary blow-up elliptic problems.
Hindawi Publishing Corporation Boundary Value Problems Volume 2006, Article ID 75674, Pages 1–10 DOI 10.1155/BVP/2006/75674
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A singular Dirichlet problem For k ≡ 0 on Ω, problem (1.1) is the following one: −Δu = g(u),
u > 0, x ∈ Ω, u|∂Ω = 0.
(1.2)
The problem was discussed and extended to the more general problems in a number of works, see, for instance, [4, 5, 7, 8, 11, 14–17]. Fulks and Maybee [7], Stuart [14], Crandall et al. [4] showed that if g satisfies (g1 ), then problem (1.2) has a unique solution u0 ∈ C 2+α (Ω) ∩ C(Ω). Moreover, Crandall et al. [4, Theorems 2.2 and 2.7] showed that there exist positive constants C1 and C2 such that (I) C1 ψ(d(x)) ≤ u0 (x) ≤ C2 ψ(d(x)) near ∂Ω, where d(x) = dist(x,∂Ω), ψ ∈ C[0,a] ∩ C 2 (0,a] is the local solution to the problem −ψ (s) = g ψ(s) ,
ψ(s) > 0,
0 < s < a,
ψ(0) = 0.
(1.3)
Then, for g(u) = u−γ , γ > 0, Lazer and McKenna [11], by construction of the global subsolution and supersolution, showed that u0 has the following properties: (I1 ) if γ > 1, then C1 [φ1 (x)]2/(1+γ) ≤ u0 (x) ≤ C2 [φ1 (x)]2/(1+γ) on Ω; / C 1 (Ω); (I2 ) if γ > 1, then u0 ∈ 1 (I3 ) u0 ∈ H0 (Ω) if and only if γ < 3, this is a basic character to
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