Abstract
By Karamata regular variation theory, we show the existence and exact asymptotic behaviour of the unique classical solution near the boundary to a singular Dirichlet problem , , , , where is a bounded domain with smooth boundary in , , , for each and some ; and for some , which is nonnegative on and may be unbounded or singular on the boundary.
Highlights
Introduction and the main resultsThe 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)where Ω is a bounded domain with smooth boundary in RN (N ≥ 1), k ∈ Clαoc(Ω) for some α ∈ (0,1), which is nonnegative on Ω, and g satisfies (g1) g ∈ C1((0, ∞), (0, ∞)), g (s) ≤ 0 for all s > 0, lims→0+ g(s) = +∞
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, ∞)), limt→0+ (g(ξt)/g(t)) = ξ−γ, for each ξ > 0 and some γ > 1; and k ∈ Clαoc(Ω) for some α ∈ (0, 1), which is nonnegative on Ω and may be unbounded or singular on the boundary
Where Ω is a bounded domain with smooth boundary in RN (N ≥ 1), k ∈ Clαoc(Ω) for some α ∈ (0,1), which is nonnegative on Ω, and g satisfies (g1) g ∈ C1((0, ∞), (0, ∞)), g (s) ≤ 0 for all s > 0, lims→0+ g(s) = +∞
Summary
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:. 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 Dıaz and Letelier [6], Lasry and Lions [10] for boundary blow-up elliptic problems.
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