Nonexistence of positive supersolutions to a class of semilinear elliptic equations and systems in an exterior domain

In this paper,we consider the following semilinear elliptic equation:■whereΩis an exterior domain in R^N with N≥3,h:Ω×R^+→R is a measurable function,and derive optimal nonexistence results of positive supersolutions.Our argument is based on a nonexistence result of positive supersolutions of a linear elliptic problem with Hardy potential.We also establish sharp nonexistence results of positive supersolutions to an elliptic system.

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 Δu = k（x）g（u）, u 〉 0, x ∈ Ω, u｜δΩ =＋∞, where Ω is a bounded domain with smooth boundary in R^N; g ∈ C^1[0, ∞）, g（0） = g＇（0） = 0, and there exists p 〉 1, such that lim g（sξ）/g（s）=ξ^p, ↓Aξ 〉 0, and k ∈ Cloc^α（Ω） is non-negative non-trivial in D which may be singular on the boundary.

Existence of Explosive Solutions for a Class of Nonlinear Elliptic Equations

By the subsuper solutions method, the explosive supersolutions and explosive subsol utions are obtained and the exsistence of explosive solutions is proved on a bounded domain for a class of nonlinear elliptic problems.Then, the exsitence of an entire large solution is proved by the perturbed method.

POSITIVE EQUILIBRIUM SOLUTIONS OF SEMILINEAR PARABOLIC EQUATIONS

The author studies semilinear parabolic equations with initial and periodic boundary value conditions. In the presence of non-well-ordered sub- and super-solutions： ＂subsolution ≮ supersolution＇, the existence and stability/instability of equilibrium solutions are obtained.

STRUCTURE OF NONNEGATIVE NONTRIVIAL AND POSITIVE SOLUTIONS OF SINGULARLY PERTURBED p-LAPLACE EQUATIONS

Structure of nonnegative nontrivial and positive solutions was precisely studied for some singularly perturbed p-Laplace equations. By virtue of sub- and supersolution method, it is shown that there are many nonnegative nontrivial spike-layer solutions and positive intermediate spike-layer solutions. Moreover, the upper and lower bound on the measure of each spike-layer were estimated when the parameter is sufficiently small.

THE EXISTENCE AND UNIQUENESS OF THE DECAYING POSITIVE ENTIRE SOLUTIONS FOR A CLASS OF SEMILIN EARELLIPTIC EQUATIONS

This paper first proves the following equations△u-m￣2u+f(x,u)=0, x(R￣n,n≥3 m>0 existence of decaying positive entire solution, then emphatically, proves this solution'suniqueness.

STRUCTURE OF POSITIVE SOLUTIONS OF A CLASS OF QUASILINEAR ORDINARY DIFFERENTIAL EQUATIONS

The existence and uniqueness of positive solutions for a class of quasilinear ordinary differential equations with a large parameter are obtained. It is shown that the flat core of positive solutions can exist. So some results of [1-3] are sharpened.

Existence and Stability of Positive Solutions for an Elliptic Cooperative System

By discussing the properties of a linear cooperative system, the necessary and sufficient conditions for the existence of positive solutions of an elliptic cooperative system in terms of the principal eigenvalue of the associated linear system are established, and some local stability results for the positive solutions are also obtained.

EXISTENCE AND NONEXISTENCE OF GLOBAL SOLUTION OF NONLINEAR PARABOLIC EQUATION WITH NONLINEAR BOUNDARY CONDITION

This paper deals with the existence and nonexistence of global positive solution of the following equation:where p, q, m, α are parameters with is a bounded domain with Ω smooth enough, The necessary and sufficient conditions for the global existence of solution are obtained.

Nonlinear Doob—Meyer Decomposition with Jumps

Concepts of g-supersolution, g-manrtingale, g-supermartingale are introduced, which are related to BSDE with Brownian motion and Poisson point process. A strict comparison theorem, monotonic limit theorem related to this type of BSDE are also discussed. As an application of these results, a nonlinear Doob-Meyer decomposition theorem is obtained.

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy-Sobolev-Maz＇ya term：-Δu-λ u/｜y｜2 = （｜u｜pt-1u）/｜y｜t ＋ μf（x）, x∈ Ω,where Ω is a bounded domain in RN（N ≥ 2）, 0 ∈ Ω, x = （y, z） ∈ Rk ×RN-k and pt = （N＋2-2t）/（N-2） （0 ≤ t ≤ 2）. For f（x） ∈ C1（Ω）／{0}, we show that there exists a constant μ＊ 〉0 such that the problem possesses at least two positive solutions if μ ∈ （0, μ＊） and at least one positive solution if μ = μ＊. Furthermore, there are no positive solutions if μ ∈ （μ＊,＋∞）.

Increasing Powers in a Degenerate Parabolic Logistic Equation

The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problem tu- △u=au-b（x）up in Ω×R＋,u（0）=u0,u（t ）｜ Ω=0, as p→ ＋∞, where Ω is a bounded domain, and b（x） is a nonnegative function. The authors deduce that the limiting configuration solves a parabolic obstacle problem, and afterwards fully describe its long time behavior.

Existence and Asymptotic Behavior of Boundary Blow-Up Weak Solutions for Problems Involving the p-Laplacian

Let D C RN （N≥3）, be a smooth bounded domain with smooth boundary 3D. In this paper, the existence of boundary blow-up weak solutions for the quasilinear elliptic equation Δpu -= Ak（x）f（u） in D（λ 〉 0 and 1 〈 p 〈 N）, is obtained under new conditions on k. We give also asymptotic behavior near the boundary of such solutions.