Bifurcation for strongly indefinite functional and applications to Hamiltonian system and noncooperative elliptic system

In this paper, we discuss the bifurcation problems for strongly indefinite functional via Morse theory. The generalized topological degree for a class of vector fields is defined. As applications, we study the bifurcation problems for Hamiltonian system and noncooperative elliptic system.     

Computations of critical groups and applications to asymptotically linear wave equation and beam equation

In this paper, by using the Morse index theory for strongly indefinite functionals developed in [Nonlinear Anal. TMA, in press], we compute precisely the critical groups at the origin and at infinity, respectively. The abstract theorems are used to study the existence (multiplicity) of nontrivial periodical solutions for asymptotically wave equation and beam equation with resonance both at infinity and at zero.     

Nontrivial solutions for resonant noncooperative elliptic systems

In this paper, by introducing some new conditions, we study the nontrivial (multiple) solutions for resonant noncooperative elliptic systems. Our main ingredients are using a new version of Morse theory for strongly indefinite functionals and precisely computing the critical groups of the associated variational functionals at zero and at infinity. © 2000 John Wiley & Sons, Inc.     

Conjugate Direction Methods and Polarity for Quadratic Hypersurfaces

We use some results from polarity theory to recast several geometric properties of Conjugate Gradient-based methods, for the solution of nonsingular symmetric linear systems. This approach allows us to pursue three main theoretical objectives. First, we can provide a novel geometric perspective on the generation of conjugate directions, in the context of positive definite systems. Second, we can extend the above geometric perspective to treat the generation of conjugate directions for handling indefinite linear systems. Third, by exploiting the geometric insight suggested by polarity theory, we can easily study the possible degeneracy (pivot breakdown) of Conjugate Gradient-based methods on indefinite linear systems. In particular, we prove that the degeneracy of the standard Conjugate Gradient on nonsingular indefinite linear systems can occur only once in the execution of the Conjugate Gradient.     

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.     

On the existence and shape of least energy solutions for some elliptic systems

We establish an existence result for strongly indefinite semilinear elliptic systems with Neumann boundary condition, and we study the limiting behavior of the positive solutions of the singularly perturbed problem.     

Symmetry breaking of solutions of non-cooperative elliptic systems

In this article we study the symmetry breaking phenomenon of solutions of non-cooperative elliptic systems. We apply the degree for G$G$-invariant strongly indefinite functionals to obtain simultaneously a symmetry breaking and a global bifurcation phenomenon.     

A new morse index theory for strongly indefinite functionals

In this paper, a new Morse index theory for strongly indefinite functionals was developed via Gălerkin approximation. In particular, the abstract theory is valid for those kinds of strongly indefinite functionals corresponding to wave equation and beam equation.     

A QMR-based interior-point algorithm for solving linear programs

A new approach for the implementation of interior-point methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite 2×2-block systems of linear equations that arise within the interior-point algorithm. These linear systems are solved by a symmetric variant of the quasi-minimal residual (QMR) algorithm, which is an iterative solver for general linear systems. The symmetric QMR algorithm can be combined with indefinite preconditioners, which is crucial for the efficient solution of highly indefinite linear systems, yet it still fully exploits the symmetry of the linear systems to be solved. To support the use of the symmetric QMR iteration, a novel stable reduction of the original unsymmetric 3×3-block systems to symmetric 2×2-block systems is introduced, and a measure for a low relative accuracy for the solution of these linear systems within the interior-point algorithm is proposed. Some indefinite preconditioners are discussed. Finally, we report results of a few preliminary numerical experiments to illustrate the features of the new approach.     

强不定问题的变分方法

简要回顾近年来关于强不定问题的变分方法某些研究方面的发展．首先介绍强不定问题,接着叙述建立强不定问题的变分框架的基本思路,进而给出局部凸拓扑线性空间的形变理论,最后陈述几个基于此形变理论的处理强不定问题的临界点定理．这些理论的应用将在后续文章中介绍．     

Bourgin-Yang Type Theorem and its application to -equivariant Hamiltonian systems

We will be concerned with the existence of multiple periodic solutions of asymptotically linear Hamiltonian systems with the presence of -action. To that purpose we prove a new version of the Bourgin-Yang theorem. Using the notion of the crossing number we also introduce a new definition of the Morse index for indefinite functionals.

     

The effects of concave and convex nonlinearities in some noncooperative elliptic systems

We consider a class of elliptic systems leading to strongly indefinite functionals, with nonlinearities which involve a combination of concave and convex terms. Using variational methods, we prove the existence of infinitely many large and small energy solutions. Our approach relies on new critical point theorems which guarantee the existence of infinitely many critical values of a wide class of strongly indefinite even functionals. Our abstract critical points theorems generalize the fountain theorems of T. Bartsch and M. Willem.     

Infinitely many solutions for Hamiltonian systems

We consider two classes of the second-order Hamiltonian systems with symmetry. If the systems are asymptotically linear with resonance, we obtain infinitely many small-energy solutions by minimax technique. If the systems possess sign-changing potential, we also establish an existence theorem of infinitely many solutions by Morse theory.     

NONTRIVIAL SOLUTIONS OF COMPETITIVE-DIFFUSIVE SYSTEMS WITH SMALL DIFFUSION

ＮＯＮＴＲＩＶＩＡＬＳＯＬＵＴＩＯＮＳＯＦＣＯＭＰＥＴＩＴＩＶＥ－ＤＩＦＦＵＳＩＶＥＳＹＳＴＥＭＳＷＩＴＨ　ＳＭＡＬＬＤＩＦＦＵＳＩＯＮＬｉＤａｈｕａ（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＨｕａｚｈｏｎｇＵｎｉｖ．ｏｆＳｃｉ．＆Ｔｅｃｈ．，Ｗｕｈａｎ４３００７...     

两个符号的等长代换子系统的混沌性态

本文研究了由两个符号的等长代换生成的子系统的混沌性态．通过分类研究,我们得到：（1）非奇异代换系统不是Li－Yorke混沌的,作为特例,Morse极小系统不是混沌的;（2）奇异代换子系统都是Li－Yorke混沌的．     

Existence of multiple nontrivial solutions for a Schrödinger–Poisson system

We consider a Schrödinger–Poisson system in R³${\mathbb{R}}^3$ with potential indefinite in sign and a general nonlinearity. We use the direct variational method and Morse theory to obtain the existence of multiple nontrivial solutions for this system.     

Critical points theorems concerning strongly indefinite functionals and infinite many periodic solutions for a class of Hamiltonian systems

Based on new deformation theorems concerning strongly indefinite functionals, we give some new min-max theorems which are useful in looking for critical points of functionals which are strongly indefinite and satisfy Cerami condition instead of Palais-Smale condition. As one application of abstract results, we study existence of multiple periodic solutions for a class of non-autonomous first order Hamiltonian system

     

Standing waves to discrete vector nonlinear Schrödinger equation

The purpose of this paper is to study a class of periodic discrete vector nonlinear Schrödinger equation. By using the critical point theory for strongly indefinite problems developed by Ding (Interdisciplinary Mathematical Sciences, World Scientific, Hackensack, NJ, 2007), we prove the existence of non-trivial standing waves for the vector equation with periodic or asymptotically periodic nonlinearities.     

Equivalent Conditions for the Solvability of Nonstandard and Singular LQ-Problems with Application to Parabolic Equations

We consider an optimal control problem with indefinite cost for an abstract model, which covers, in particular, parabolic systems in a general bounded domain. Necessary and sufficient conditions are given for the synthesis of the optimal control, which is given in terms of the Riccati operator arising from a nonstandard Riccati equation. The theory extends also a finite-dimensional frequency theorem to the infinite-dimensional setting. Applications include the heat equation with Dirichlet and Neumann controls, as well as the strongly damped Euler–Bernoulli and Kirchhoff equations with the control in various boundary conditions.     

INFINITELY MANY SOLUTIONS FOR ELLIPTIC SYSTEMS WITH STRONGLY INDEFINITE VARIATIONAL STRUCTURE

Based on the multiplicity results of Benci and Fortunato [4], we consider some elliptic systems with strongly indefinite quadratic part, and establish the existence of infinitely many nontrivial solutions in a suitable family of products of fractional Sobolev spaces.