Some Abstract Critical Point Theorems for Self-adjoint Operator Equations and Applications

By using the index theory for linear bounded self-adjoint operators in a Hilbert space related to a fixed self-adjoint operator A with compact resolvent,the authors discuss the existence and multiplicity of solutions for(nonlinear) operator equations,and give some applications to some boundary value problems of first order Hamiltonian systems and second order Hamiltonian systems.     

临界情形的非齐次半线性椭圆型方程的多解性

考虑有界区域Ω RN 上非齐次半线性椭圆型方程 -Δu(x) =up(x) λf(x)在齐次混合边值条件 (即第三边值问题 ) u n au Ω =0下正解的存在性 ,其中α ,λ≥ 0 ,p=N 2N- 2 ,N>2 ,f(x) ∈L∞(Ω) .证明了存在常数λ >0 ,当λ∈ (0 ,λ )时 ,上述问题至少存在两个正解     

Nontrivial Solutions for Some Semilinear Elliptic Equations with Critical Sobolev Exponents

Let Ω be a bounded domain in R^4(n ≥ 4) with smooth boundary ∂Ω. We discuss the existence of nontrivial solutions of the Dirichlet problem {- Δu = a(x) |u|^{4/(a-2)}u + λu + g(x, u), \quad x ∈ Ω u = 0, \quad x ∈ ∂Ω where a(x) is a smooth function which is nonnegative on \overline{Ω} and positive somewhere, λ> 0 and λ ∉ σ(-Δ). We weaken the conditions on a(x) that are generally assumed in other papers dealing with this problem.     

一种奇异临界椭圆方程的非平凡解

该文研究了一类带有Sobolev-Hardy临界指数的半线性椭圆方程,运用变分理论中的环绕定理证明了方程非平凡解的存在性     

关于半线性椭圆共振问题的注记

本文应用Ｍｏｒｓｅ理论和惩罚性技巧研究了一类半线性椭圆方程在无穷远处和在原点处都共振情形下非平凡解的存在性．     

具临界Sobolev-Hardy指数的半线性椭圆方程的非平凡解

本文研究了如下问题：-div（｜x｜β△u）=｜x｜^a｜u｜^2（α,β）-2u＋λ｜x｜σ｜u｜^q-2,x∈Ω,u=0,x∈δΩ,这里Ω∪→R^N是有界光滑区域且0∈Ω，2（α，β）=2（N＋α）/N＋β-2，运用Sobolev-Hardy不等式和山路几何，证明了在一定的条件下方程至少存在一个非平凡解。     

On the Existence of Positive Solutions of Quasilinear Elliptic Equations with Mixed Boundary Conditions

In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation {-div (|∇u|^{p-2}∇u) = |u|^{p^∗-2}u + f(x, u), \quad u > 0, \quad x ∈ Ω u|_Γ_0 = 0, \frac{∂u}{∂\overrightarrow{n}}|_Γ_1 = 0 is obtained, where Ω is a bounded smooth domain in R^N, ∂Ω = \overrightarrow{Γ}_0 ∪ \overrightarrow{Γ}_1, 2 ≤ p < N, p^∗ = \frac{Np}{N-p}, Γ_0 and Γ_1 are disjoint open subsets of ∂Ω.     

Regularity and Morse Index of the Solutions to Critical Quasilinear Elliptic Systems

Regularity results and critical group estimates are studied for critical (p, r)-systems. Multiplicity results of solutions for a critical potential quasilinear system are also proved using Morse theory.     

一种几何指标理论在泛函微分方程中的应用

本文给出一种Ｚｐ群作用下不变的新几何指标,利用这种指标理论,得到一 类周期时变泛函微分方程次调和解的存在性条件．同时我们对文［１］有关Ｈａｍｉｌｔｏｎ系 统次调和解存在性的猜想给出了一个新解答．     

Tornado solutions for semilinear elliptic equations in : Applications

We show partial regularity of bounded positive solutions of some semilinear elliptic equations in domains of . As a consequence, there exists a large variety of nonnegative singular solutions to these equations. These equations have previously been studied from the point of view of free boundary problems, where solutions additionally are stable for a variational problem, which we do not assume.

     

Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations

We prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution. To this aim, we prove an appropriate fixed point theorem in partially ordered sets. ★ Partially supported by Ministerio de Ciencia y Tecnología/ FEDER, project BFM2001-3884-C02-01; and by Xunta de Galicia/ FEDER, project PGIDIT02PXIC20703PN.     

Comparison Theorems in Delay Differential Equations in a Critical State and Applications

First, two comparison theorems are established between the firstorder delay differential equation x'(t)+p(t)x((t))=0, tt₀ (*) and two related second order ordinary differential equationsin the case when Next, by using these comparison theorems some new oscillationand non-oscillation criteria are given for (*) under condition(**). As a consequence, an open problem by Elbert and Stavroulakisis completely solved.     

无穷区间上二阶脉冲微分方程多点边值问题的正解

对无穷区间上二阶脉冲微分方程多点边值问题进行了讨论,主要利用Schauder不动点定理及对角化过程研究了所述问题的正解的存在性．     

变分方法求解脉冲微分方程

本文研究一类具有周期边值条件的脉冲微分方程解的存在性问题.通过运用临界点理论以及变分方法证明此系统存在周期解.     

Existence and multiplicity results for classes of elliptic resonant problems

In this paper we prove some new existence results of nontrivial solutions for classes of elliptic resonant problems. We also establish several multiplicity results. The methods used here are based on combining the minimax methods and the Morse theory especially some new observations on the critical groups of a local linking-type degenerate critical point.     

新的不动点定理及应用

本文获得了几个压缩型映射的不动点定理,改进了相应文献的结果．应用该定理,给出了一类时滞积分方程的正概周期解的存在性结果．推广了已有的结果．     

偏序度量空间中的若干不动点定理及其在周期边值问题中的应用

本文给出了偏序度量空间中的若干不动点定理.利用这些不动点定理,我们给出了周期边值问题存在唯一解的一个充分条件.     

含临界位势与临界参数的超线性椭圆型方程

本文考虑一类含临界位势与临界参数的超线性椭圆型方程解的存在性.本文应用Morse理论,考虑非线性项f(x,s)在零点附近以及无穷远处的性质,给出了方程在某个新的Sobolev-Hardy空间中解的存在性.