Karamata Regular Variation Theory Research Articles

Second order expansion for blowup solutions of semilinear elliptic problems

In this paper, we mainly study the second order expansion of solutions to the elliptic problems △ u = b ( x ) f ( u ) , u > 0 , x ∈ Ω , u | ∂ Ω = ∞ , where Ω is a bounded domain with a smooth boundary in R N ( N ≥ 2 ) , b ∈ C α ( Ω ̄ ) which is positive in Ω and may be vanishing on the boundary and f is normalized regularly varying at infinity with index p > 1 . Our analysis is based on the sub-supersolution method and Karamata regular variation theory.

Singular solutions of perturbed logistic-type equations

We are concerned with the qualitative analysis of positive singular solutions with blow-up boundary for a class of logistic-type equations with slow diffusion and variable potential. We establish the exact blow-up rate of solutions near the boundary in terms of Karamata regular variation theory. This enables us to deduce the uniqueness of the singular solution.

Large solution to nonlinear elliptic equation with nonlinear gradient terms

The aim of this paper is to study the qualitative behavior of large solutions to the following problem { Δ u ± a ( x ) | ∇ u | q = b ( x ) f ( u ) , x ∈ Ω , u ( x ) = ∞ , x ∈ ∂ Ω . Here a ( x ) ∈ C α ( Ω ) is a positive function with α ∈ ( 0 , 1 ) , b ( x ) ∈ C α ( Ω ) is a non-negative function and may be singular near the boundary or vanish on the boundary, and the nonlinear term f is a Γ-varying function, whose variation at infinity is not regular. We focus our investigation on the existence and asymptotic behavior close to the boundary ∂ Ω of large solutions by Karamata regular variation theory and the method of upper and lower solution. The main results of this paper emphasize the central role played by the gradient term | ∇ u | q and the weight functions a ( x ) and b ( x ) .

Boundary behaviour of explosive solution to quasilinear elliptic problems with nonlinear gradient terms

Based on the Karamata regular variation theory and the method of explosive sub and supersolution, the boundary behaviour of explosive solutions to the quasilinear elliptic equation was obtained, where the singular weight function is non-negative and non-trivial, which may be unbounded on the boundary, the nonlinear term is a Γ-varying function, whose variation at infinity is not regular. The results of this article emphasize the central role played by the gradient term and singular weight function.

A second-order estimate for blow-up solutions of elliptic equations

We investigate second-term asymptotic behavior of boundary blow-up solutions to the problems Δ u = b ( x ) f ( u ) , x ∈ Ω , subject to the singular boundary condition u ( x ) = ∞ , in a bounded smooth domain Ω ⊂ R N . b ( x ) is a non-negative weight function. The nonlinearly f is regularly varying at infinity with index ρ > 1 (that is lim u → ∞ f ( ξ u ) / f ( u ) = ξ ρ for every ξ > 0 ) and the mapping f ( u ) / u is increasing on ( 0 , + ∞ ) . The main results show how the mean curvature of the boundary ∂ Ω appears in the asymptotic expansion of the solution u ( x ) . Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.

Second order estimates for large solutions of elliptic equations

This paper deals with the second term asymptotic behavior of large solutions to the problems Δ u = b ( x ) f ( u ) , x ∈ Ω , subject to the singular boundary condition u ( x ) = ∞ , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R N , and b ( x ) is a non-negative weight function. The absorption term f is regularly varying at infinite with index ρ > 1 (that is lim u → ∞ f ( ξ u ) / f ( u ) = ξ ρ for every ξ > 0 ) and the mapping f ( u ) / u is increasing on ( 0 , + ∞ ) . Our analysis relies on the Karamata regular variation theory.

Boundary behavior of solutions for the degenerate logistic type elliptic problem with boundary blow-up

By the Karamata regular variation theory and constructing comparison function, we show the exact asymptotic behavior of solutions for the degenerate logistic type elliptic problem with boundary blow-up.

Boundary blow-up rates of large solutions for elliptic equations with convection terms

By Karamata regular variation theory, a perturbation method and constructing comparison functions, we show the exact asymptotic behavior of large solutions to the semilinear elliptic equations with convection terms { Δ u ± | ∇ u | q = b ( x ) f ( u ) , x ∈ Ω , u ( x ) = + ∞ , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R N . The weight function b ( x ) is a non-negative continuous function in the domain, which may be vanishing on the boundary or be singular on the boundary. f ( u ) ∈ C 2 [ 0 , + ∞ ) is increasing on ( 0 , ∞ ) satisfying the Keller–Osserman condition, and regularly varying at infinity with index ρ > 1 .

Asymptotic behavior of large solution for boundary blowup problems with non-linear gradient terms

Let Ω ⊂ R N ( N ⩾ 3 ) be a bounded domain with smooth boundary. We show the asymptotic behavior of boundary blowup solutions to non-linear elliptic equation Δ u ± | ∇ u | q = b ( x ) f ( u ) in Ω , subject to the singular boundary condition u ( x ) = ∞ as dist ( x , ∂ Ω ) → 0 , f is Γ -varying at ∞ . Our analysis is based on the Karamata regular variation theory combined with the method of lower and supper solution.

Asymptotic behavior of large solutions to [formula omitted]-Laplacian of Bieberbach–Rademacher type

By the Karamata regular variation theory and the method of lower and upper solutions, we establish the asymptotic behavior of boundary blow-up solutions of the quasilinear elliptic equation div ( | ▽ u | p − 2 ▽ u ) = b ( x ) f ( u ) in a bounded Ω ⊂ R N subject to the singular boundary condition u ( x ) = ∞ , where the weight b ( x ) is non-negative and non-trivial in Ω , which may be vanishing on the boundary or go to unbounded, the nonlinear term f is a Γ -varying function at infinity, whose variation at infinity is not regular.

Boundary blow-up elliptic problems with nonlinear gradient terms and singular weights

By Karamata regular variation theory, a perturbation method and construction of comparison functions, we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems Δu ± |Δu|q = b(x)g(u), u > 0 in Ω, u|∂Ω = ∞, where Ω is a bounded domain with smooth boundary in ℝN, q > 0, g ∈ C1[0, ∞) is increasing on [0, ∞), g(0) = 0, g′ is regularly varying at infinity with positive index ρ and b is non-negative in Ω and is singular on the boundary.

The existence and global optimal asymptotic behaviour of large solutions for a semilinear elliptic problem

By Karamata regular variation theory and constructing comparison functions, the author shows the existence and global optimal asymptotic behaviour of solutions for a semilinear elliptic problem Δ u = k( x) g( u), u > 0, x ∈ ω, u|δ ω = +∞, where ω is a bounded domain with smooth boundary in ▪, and there exists p > 1, such that lim s → ∞ g ( S ξ ) g ( s ) = ξ p , ∀ ξ > 0 , a n d k ∈ C l o c α ( Ω ) is non-negative non-trivial in Ω which may be singular on the boundary.

Asymptotic Behavior of The Unique Solution to a Singular Elliptic Problem With Nonlinear Convection Term And Singular Weight

Abstract By Karamata regular variation theory, we first derived the exact asymptotic behavior of the local solution to the problem -φʹʹ(s) = g(φ(s)), φ(s) > 0, s ∈ (0, a) and φ(0) = 0. Then, by a perturbation method and constructing comparison functions, we derived the exact asymptotic behavior of the unique classical solution near the boundary to a singular Dirichlet problem -Δu = b(x)g(u) + λ|▽u|q, u > 0, x ∈ Ω, u|aΩ = 0, where Ω is a bounded domain with smooth boundary, λ ∈ ℝ, q ∈ [0, 2]; g ∈ C1((0, ∞), (0, ∞)), is decreasing in (0, ∞) with lims→0 + g(s) = +∞; the weight b is positive in Ω and singular on the boundary.

Boundary behaviour of the unique solution to a singular Dirichlet problem with a convection term

By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem − Δ u = b ( x ) g ( u ) + λ | ∇ u | q , u > 0 , x ∈ Ω , u | ∂ Ω = 0 , which is independent of λ | ∇ u λ | q , where Ω is a bounded domain with smooth boundary in R N , λ ∈ R , q ∈ ( 0 , 2 ] , lim s → 0 + g ( s ) = + ∞ , and b is non-negative on Ω, which may be vanishing on the boundary.

Boundary blow-up elliptic problems with nonlinear gradient terms

By Karamata regular variation theory and perturbation method, we show the exact asymptotical behaviour of solutions near the boundary to nonlinear elliptic problems Δ u ± | ∇ u | q = b ( x ) g ( u ) , u > 0 in Ω, u | ∂ Ω = + ∞ , where Ω is a bounded domain with smooth boundary in R N , q ⩾ 0 , g ∈ C 1 [ 0 , ∞ ) , g ( 0 ) = 0 , g ′ is regularly varying at infinity with index ρ with ρ > 0 and b is nonnegative nontrivial in Ω, which may be vanishing on the boundary.

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

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.

Asymptotics for the blow-up boundary solution of the logistic equation with absorption

Let Ω be a smooth bounded domain in R N . Assume that f⩾0 is a C 1-function on [0,∞) such that f( u)/ u is increasing on (0,+∞). Let a be a real number and let b⩾0, b≢0 be a continuous function such that b≡0 on ∂Ω. The purpose of this Note is to establish the asymptotic behaviour of the unique positive solution of the logistic problem Δ u+ au= b( x) f( u) in Ω, subject to the singular boundary condition u( x)→+∞ as dist(x,∂Ω)→0 . Our analysis is based on the Karamata regular variation theory. To cite this article: F.-C. Cîrstea, V. Rădulescu, C. R. Acad. Sci. Paris, Ser. I 336 (2003).