scholarly journals The exact asymptotic behaviour of the unique solution to a singular Dirichlet problem

2006 ◽  
Vol 2006 ◽  
pp. 1-10 ◽  
Author(s):  
Zhijun Zhang ◽  
Jianning Yu
Keyword(s):  
Dirichlet Problem ◽  
Asymptotic Behaviour ◽  
Unique Solution ◽  
Singular Dirichlet Problem

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

2006 ◽  
Vol 136 (1) ◽  
pp. 209-222 ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Dirichlet Problem ◽  
Asymptotic Behaviour ◽  
Unique Solution ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Convection Term ◽  
Nonlinear Dirichlet Problem ◽  
Singular Dirichlet Problem

We show the existence and exact asymptotic behaviour of the unique solution u ∈ C2(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) = +∞, k ∈ Cα(Ω) is non-negative non-trivial on Ω, which may be singular on the boundary.

Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

2018 ◽  
Vol 18 (2) ◽  
pp. 289-302
Author(s):  
Zhijun Zhang
Keyword(s):  
Dirichlet Problem ◽  
Crucial Role ◽  
Blow Up ◽  
Boundary Behavior ◽  
Structure Condition ◽  
Bounded Smooth Domain ◽  
Convex Solution ◽  
Bounded Smooth ◽  
Monge Ampere ◽  
Singular Dirichlet Problem

AbstractThis paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation\operatorname{det}D^{2}u=b(x)g(-u),\quad u<0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0,where Ω is a strictly convex and bounded smooth domain in{\mathbb{R}^{N}}, with{N\geq 2},{g\in C^{1}((0,\infty),(0,\infty))}is decreasing in{(0,\infty)}and satisfies{\lim_{s\rightarrow 0^{+}}g(s)=\infty}, and{b\in C^{\infty}(\Omega)}is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition ongwhich plays a crucial role in the boundary behavior of such solution.

Optimal global asymptotic behaviour of the solution to a class of singular Dirichlet problems

2020 ◽  
pp. 1-19
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behaviour ◽  
Unique Solution ◽  
Blow Up ◽  
Critical Values ◽  
Smooth Domain ◽  
Dirichlet Problems ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems − Δu = b(x)g(u), u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded smooth domain in ℝ n , g ∈ C1(0, ∞) is positive and decreasing in (0, ∞) with $\lim _{s\rightarrow 0^+}g(s)=\infty$ , b ∈ Cα(Ω) for some α ∈ (0, 1), which is positive in Ω, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values.

scholarly journals Singular Dirichlet Boundary Value Problem for Second Order Ode

2007 ◽  
Vol 14 (2) ◽  
pp. 325-340
Author(s):  
Irena Rachůnková ◽  
Jakub Stryja
Keyword(s):  
Dirichlet Problem ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Value Problem ◽  
Existence Of A Solution ◽  
Absolutely Continuous ◽  
First Derivative ◽  
Carathéodory Conditions ◽  
Time Singularities ◽  
Existence Principle ◽  
Singular Dirichlet Problem

Abstract This paper investigates the singular Dirichlet problem –𝑢″ = 𝑓(𝑡, 𝑢, 𝑢′), 𝑢(0) = 0, 𝑢(𝑇) = 0, where 𝑓 satisfies the Carathéodory conditions on the set and . The function 𝑓(𝑡, 𝑥, 𝑦) can have time singularities at 𝑡 = 0 and 𝑡 = 𝑇 and space singularities at 𝑥 = 0 and 𝑦 = 0. The existence principle for the above problem is given and its application is presented here. The paper provides conditions which guarantee the existence of a solution which is positive on (0; T) and which has the absolutely continuous first derivative on [0, 𝑇].

scholarly journals On a problem of lower limit in the study of nonresonance

1997 ◽  
Vol 2 (3-4) ◽  
pp. 227-237
Author(s):  
A. Anane ◽  
O. Chakrone
Keyword(s):  
Dirichlet Problem ◽  
Asymptotic Behaviour ◽  
Lower Limit ◽  
First Eigenvalue ◽  
The First Eigenvalue

We prove the solvability of the Dirichlet problem{−Δpu=f(u)+h   in   Ω,              u=0                     on  ∂Ωfor every givenh, under a condition involving only the asymptotic behaviour of the potentialFoffwith respect to the first eigenvalue of thep-LaplacianΔp. More general operators are also considered.

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