Morse Index of Multiple Blow-up Solutions to the Two-Dimensional Sinh-Poisson Equation

In this paper we consider the Dirichlet problemwhere $\rho$ is a small parameter and $\Omega$ is a $C^2$ bounded domain in $\mathbb{R}^2$. In [1], the author proves the existence of a $m$-point blow-up solution $u_\rho$ jointly with its asymptotic behaviour. We compute the Morse index of $u_\rho$ in terms of the Morse index of the associated Hamilton function of this problem. In addition, we give an asymptotic estimate for the first $4m$ eigenvalues and eigenfunctions.     

具有多重解的非线性奇摄动问题

利用边界层法,研究了一类具有多重解的非线性奇摄动问题．在适当的假设下,通过给出外部解展开式系数及其对应边界条件的一般表达式,根据退化问题的边值作为某方程的根的重数,得到了此问题不同形式的渐近解．特别地,当这种根的重数为偶数时,问题具有二重解．另外,将相关结果应用于化学反应器理论,并通过对具有多重解的例子的渐近解和精确解的数值模拟说明如此构造的渐近解具有较高的精度．     

Homotopical linking and Morse index estimates in min-max theorems

The authors extend some well-known Morse estimates for critical points of saddle point type to some linking conditions recently considered in the literature. Applications are given for multiplicity results in PDE and existence of subharmonic solutions for a class of conservative ODE. Research supported by Program STRIDE (contract STRDA/C/CEN/531/92) and EC (contract ERBCHRXCT940555).     

A Morse Index Theorem for Elliptic Operators on Bounded Domains

Given a selfadjoint, elliptic operator L, one would like to know how the spectrum changes as the spatial domain Ω ? ? ⁿ is deformed. For a family of domains {Ω _t } _{t∈[a, b]} we prove that the Morse index of L on Ω _a differs from the Morse index of L on Ω _b by the Maslov index of a path of Lagrangian subspaces on the boundary of Ω. This is particularly useful when Ω _a is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the difference of Morse indices for the “original” problem (on Ω _b ) and the “simplified” problem (on Ω _a ). This generalizes previous multi-dimensional Morse index theorems that were only available on star-shaped domains or for Dirichlet boundary conditions. We also discuss how one can compute the Maslov index using crossing forms, and present some applications to the spectral theory of Dirichlet and Neumann boundary value problems.     

非线性三阶微分方程组边值问题的多个正解

本文通过利用范数形式的锥拉伸和压缩不动点定理及Leggett-Williams不动点定理,获得了非线性三阶微分方程组边值问题多个正解的存在性,并给出了一些例子说明结果的应用.     

Multiple Solutions for a Fourth-order Asymptotically Linear Elliptic Problem

Under simple conditions, we prove the existence of three solutions for a fourth-order asymptotically linear elliptic boundary value problem. For the resonance case at infinity, we do not need to assume any more conditions to ensure the boundedness of the （PS） sequence of the corresponding functional.     

Finite Morse index solutions of exponential problems

We prove that the problem −Δu=eu$-\mathrm{\Delta }u={\mathrm{e}}^u$ has no negative finite Morse index solution on R³${\mathbb{R}}^3$ and give some applications to bounded domain problems.     

关于运输问题最优解的进一步讨论

首先给出了运输问题最优解的相关概念,将最优解扩展到广义范畴,提出狭义多重最优解和广义多重最优解的概念及其区别.然后给出了惟一最优解、多重最优解、广义有限多重最优解、广义无限多重最优解的判定定理及其证明过程.最后推导出了狭义有限多重最优解个数下限和广义有限多重最优解个数上限的计算公式,并举例验证了结论的正确性.     

一类奇异非线性三点边值问题的多重正解

利用锥映射的不动点指数定理,建立了一类奇异三点边值问题多个正解的存在性定理.改进和推广了相关结果.     

一类非线性二阶微分方程无穷边值问题的多重正无界解

本文通过构造—个特殊的锥,利用锥拉压不动点定理,证明了一类非线性二阶微分方程无穷边值问题的两个正无界解的存在性。     

The Morse Index of Wente Tori

We find various lower and upper bounds for the index of Wente tori that contain a continuous family of planar principal curves. We then prove a result that gives an algorithm for computing the index sharply.     

非线性二阶时滞微分方程边值问题的正解

本文讨论了一类二阶时滞微分方程边值问题的正解存在性,在不要求非线性项取值恒为非负的情形下,利用锥上不动点指数的计算得到了该问题的正解.     

具有不定位势和凹非线性项的Robin问题的解的多重性

研究非线性项由凹项λ|u|q-2(其中1＜q＜2)和在无穷远处以超线性或渐近线性增长的连续项f(x,u)组成的半线性不定位势的Robin问题,在不同情形下,运用变分法、截断技巧和Morse理论,得到了该问题的多重解的存在性结果.     

非线性二阶奇异半正微分方程组非局部边值问题的多个正解

研究利用Leggett-Williams不动点定理和平移变换,讨论了非线性二阶奇异半正微分方程组非局部边值问题三个正解的存在性.文中的主要结果推广了以前相应的工作.     

一个带Hardy项的椭圆方程的多重解

通过对偶变分方法证明了一个带Hardy项和临界非线性的非线性椭圆方程的非平凡解的存在性和多重性.     

Periodic Solutions of a Nonlinear Evolution Problem

In this paper we prove existence of periodic solutions to a nonlinear evolution system of second order partial differential equations involving the pseudo-Laplacian operator. To show the existence of periodic solutions we use Faedo-Galerkin method with a Schauder fixed point argument.     

一维$P$-Laplace二阶差分系统奇异边值问题正解的存在性

应用锥压缩锥拉伸不动点定理和Leray-Schauder 抉择定理研究了一类具有P-Laplace算子的奇异离散边值问题$$\left\{\begin{array}{l}\Delta[\phi (\Delta x(i-1))]+ q_{1}(i)f_{1}(i,x(i),y(i))=0, ~~~i\in \{1,2,...,T\}\\\Delta[\phi (\Delta y(i-1))]+ q_{2}(i)f_{2}(i,x(i),y(i))=0,\\x(0)=x(T+1)=y(0)=y(T+1)=0,\end{array}\right.$$的单一和多重正解的存在性,其中$\phi(s) = |s|^{p-2}s, ~p>1$,非线性项$f_{k}(i,x,y)(k=1,2)$在$(x,y)=(0,0)$具有奇性.