Asymptotic Behavior Of Large Solutions Research Articles

Optimal global and boundary asymptotic behavior of large solutions to the Monge-Ampère equation

This paper is mainly concerned with optimal global and boundary asymptotic behavior of strict convex large solutions to the Monge-Ampère equation detD2u=b(x)f(u), x∈Ω, where Ω is a strict convex and bounded smooth domain in Rn with n≥2, f∈C1[0,∞) (or f∈C1(R)), which is increasing in [0,∞) (or R) and satisfies the Keller-Osserman type condition, b∈C∞(Ω) is positive in Ω, but may vanish or blow up on the boundary properly. We find new structure conditions on f which play a crucial role in both global and boundary behavior of such solutions. Moreover, we reveal asymptotic behavior of such solutions when the parameters on b tend to the corresponding critical values. In addition, when f does not satisfy the Keller-Osserman type condition and Ω is a ball, we supply a necessary and sufficient condition on b for the existence of an infinitude of strict convex radially symmetric positive solutions to such problem.

Existence and boundary asymptotic behavior of large solutions of Hessian equations

In this paper, we establish the existence of large solutions of Hessian equations and obtain a new boundary asymptotic behavior of solutions.

Existence and uniqueness of large solutions for a class of non-uniformly elliptic semilinear equations

In this paper, we study existence, uniqueness, and asymptotic behavior of large solutions of second-order degenerate elliptic semilinear problems in non-divergence form. The main particularity of the problem is the interior uniform ellipticity of the equation, which degenerates on the boundary, involving an effect on the boundary blow-up profile of the solution.

Global asymptotic behavior of large solutions for a class of semilinear elliptic problems

By using Karamata regular variation theory and upper and lower solution method, we investigate the existence and the global asymptotic behavior of large solutions to a class of semilinear elliptic equations with nonlinear convection terms. In our study, the weight and nonlinearity are controlled by some regularly varying functions or rapid functions, which is very different from the conditions of previous contexts. Our results largely extend the previous works, and prove that the nonlinear convection terms do not affect the global asymptotic behavior of classical solutions when the index of the convection terms change in a certain range.

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

We use Karamata regular variation theory to study the exact asymptotic behavior of large solutions near the boundary to a class of quasilinear elliptic equations with convection terms

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 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 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.

On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in $R^n$

In this paper, we give an existence result for nonradial large solutions of the semilinear elliptic equation $\Delta u =p(x)f(u)$ in $R^N (N\ge 3)$, where $f$ is assumed to satisfy $(f_1)$ and $(f_2)$ below. The asymptotic behavior of the large solutions at infinity are also studied in the sublinear case that $f(u)$ behaves like $u^{\gamma}$ at $\infty$ for $\gamma \in (0, 1)$.

Asymptotic behavior of large solutions to H-systems with perturbations

Let Ω⊂ R 2 be a bounded smooth domain. In this paper, we study the existence and the asymptotic behavior of solutions to the Dirichlet problem of H-systems with perturbations: (P ε) Δu=2u x 1 ∧u x 2 −εu in Ω, u| ∂Ω=0, where u=(u 1,u 2,u 3)∈H 0 1(Ω; R 3) and ε∈ R . We prove that for ε∈(0,λ 1) (λ 1 denotes the 1st eigenvalue of − Δ acting on H 0 1(Ω; R 3) ), ( P ε ) admits at least one solution u ̄ ε≢0 , which blows up at exactly one point in Ω as ε→0. We also characterize the location of blow up point as the minimum point of a certain function defined on Ω, and give the exact blow up rate of || ∇ u ̄ ε|| L ∞(Ω) as ε→0.

On the asymptotic analysis of $H$-systems. I. Asymptotic behavior of large solutions

Let $\Omega\subset\mathbb R^2$ be a bounded domain, $\gamma\in C^{3,\alpha}(\partial\Omega;\mathbb R^3 )$ $(0 <\alpha <1)$ and $H>0$. We consider the asymptotic behavior of large solutions of an $H$-system $$\Delta u=2Hu_{x_1}\wedge u_{x_2}\quad\text{in $\Omega$}, \qquad u=\gamma\quad \text{on $\partial\Omega$}$$ as $H\to 0$ or as $\gamma\to 0$. We show that large solutions blow up at exactly one point in $\Omega$. The exact blow-up rate and location of blow-up point are studied.