一类半线性椭圆方程正解的唯一性

本文应用延拓方法证明了一类半线性椭圆方程正解的唯一性,改进了已有结果．     

Uniqueness of Positive Solutions of Semilinear Elliptic Equations and Related Eigenvalue Problems

In this paper we survey some recent results about the uniqueness of the solution of some semilinear elliptic Dirichlet problems in bounded domains. The presentation aims to emphasize the role of the geometrical properties of the second eigenfunction of the linearized problem in the study of the above question. This motivates the analysis of the asymptotic behaviour of these eigenfunctions and of the relative eigenvalues when the nonlinear term is a power with exponent close to the critical Sobolev exponent. Research supported by MIUR, project “Variational Methods and Nonlinear Differential Equations”. Lecture held in the Seminario Matematico e Fisico on January 31, 2005 Received: June 2005     

R~n上一类半线性椭圆方程解的存在唯一性和渐近性质

用上下解方法和位势估计，研究Rn上具有次线性项加超线性项半线性椭圆方程给出了其有界正解的存在性、唯一性和渐近性质，其中为常数，参数．     

General Uniqueness Results and Variation Speed for Blow-Up Solutions of Elliptic Equations

Let be a smooth bounded domain in R^N. We prove general uniquenessresults for equations of the form – u = au – b(x)f(u) in , subject to u = on . Our uniqueness theorem is establishedin a setting involving Karamata's theory on regularly varyingfunctions, which is used to relate the blow-up behavior of u(x)with f(u) and b(x), where b 0 on and a certain ratio involvingb is bounded near . A key step in our proof of uniqueness usesa modification of an iteration technique due to Safonov. 2000Mathematics Subject Classification 35J25 (primary), 35B40, 35J60(secondary).     

一类半线性椭圆型方程衰减正整解的存在唯一性

本文首先证明方程△u-m2u+f(x,u)=0,x∈Rn,n≥3,m>0存在衰减的正整解,然后重点证明这种解的唯一性。     

半线性椭圆方程的正解

本文用特征理论及上下解方法,证明了一类半线性椭圆方程边值问题的正解的存在性,同时给出了解的估计.     

Quantitative Local Bounds for Subcritical Semilinear Elliptic Equations

The purpose of this paper is to prove local upper and lower bounds for weak solutions of semilinear elliptic equations of the form ???u =?cu ^p , with 0 < p < p _s = (d + 2)/(d - 2), defined on bounded domains of ${{\mathbb{R}^d}, d \geq 3}$ , without reference to the boundary behaviour. We give an explicit expression for all the involved constants. As a consequence, we obtain local Harnack inequalities with explicit constants, as well as gradient bounds.     

一类半线性椭圆方程的正径向解

本文以关于非线性全连续算子的锥不动点定理为工具，研究半线性椭圆边值问题上Δu λa(|x|)u f(|x|,u)=0(x∈Ω)，u|=0及Δu λf(|x|,u)=0(x∈Ω)，u|=0．在不假定f单调的情况下，本文得出了上述问题存在正径向解的若干充分条件．     

非局部奇摄动半线性椭圆型方程(英文)

本文讨论了半线性椭圆型方程非局部奇摄动问题 .在适当条件下 ,利用不动点原理 ,对边值问题解的存在性和渐近性态作了研究 .     

Existence Theorems for Certain Elliptic and Parabolic Semilinear Equations

Existence theorems for the nonlinear parabolic differential equation −∂u/∂t + Δu + |u|^p + f(x, t) = 0 in ⁿ × [0, ∞) with zero initial value are established given explicit conditions on the nonhomogeneous termf(x, t). An existence theorem is also demonstrated for the corresponding elliptic equation.     

Degenerate Semilinear Elliptic Equations with Critical Exponent

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一类半线性椭圆方程的概率解法(英)

设D为Rd（d≥3）内的无界区域，本文以超布朗运动为工具证明了下述这值问题解的存在性和唯一性．     

Solutions of Semilinear Elliptic Equations in Tubes

Given a smooth compact k-dimensional manifold Λ embedded in ? ^m , with m≥2 and 1≤k≤m?1, and given ?>0, we define B _? (Λ) to be the geodesic tubular neighborhood of radius ? about Λ. In this paper, we construct positive solutions of the semilinear elliptic equation $$\begin{cases} \Delta u + u^p = 0 &\mbox{in } B_{\epsilon}(\varLambda) \\ u = 0 & \mbox{on } \partial B_{\epsilon}(\varLambda) , \end{cases} $$ when the parameter ? is chosen small enough. In this equation, the exponent p satisfies either p>1 when n:=m?k≤2 or $p\in(1, \frac{n+2}{n-2})$ when n>2. In particular, p can be critical or supercritical in dimension m≥3. As ? tends to 0, the solutions we construct have Morse index tending to infinity. Moreover, using a Pohozaev type argument, we prove that our result is sharp in the sense that there are no positive solutions for $p>\frac{n+2}{n-2}$ , n≥3, if ? is sufficiently small.     

Multiple Solutions for a Class of Semilinear Elliptic Equations

We show that for a class of semilinear elliptic equations there are at least three nontrivial solutions. Existence of infinitely many solutions is also shown when the nonlinear term is odd. In our results, the nonlinear term can grow super-critically at infinity.

     

一种半线性椭圆型方程的解的可去奇点

本文处理了一种半线性椭圆型方程解的可去奇点问题,得到方程的弱解与一个定义在RN上的连续函数几乎处处相等的结论.