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

     

Existence results for solutions to nonlinear Dirac equations on compact spin manifolds

We consider super-linear and sub-linear nonlinear Dirac equations on compact spin manifolds. Their solutions are obtained as critical points of certain strongly indefinite functionals on a Hilbert space. For both cases, we establish existence results via Galerkin type approximations and linking arguments. For a particular case of odd nonlinearities, we prove the existence of infinitely many solutions.     

Existence of solutions for a system of diffusion equations with spectrum point zero

In this paper, we consider the existence and multiplicity of homoclinic type solutions to a system of diffusion equations with spectrum point zero. By using some recent critical point theorems for strongly indefinite problems, we obtain at least one nontrivial solution and also infinitely many solutions.     

Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first-order nonperiodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we give some new criteria to guarantee that Hamiltonian systems with asymptotically quadratic terms and spectrum point zero have at least one and a finite number of pairs of homoclinic orbits under some adequate conditions, respectively.     

Deformation in locally convex topological linear spaces

We are concerned with a deformation theory in locally convex topological linear spaces. A special "nice" partition of unity is given. This enables us to construct certain vector fields which are locally Lipschitz continuous with respect to the locally convex topology. The existence, uniqueness and continuous dependence of flows associated to the vector fields are established. Deformations related to strongly indefinite functionals are then obtained. Finally, as applications, we prove some abstract critical point theorems.     

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.     

A boundary value problem associated to non-homogeneous operators in Orlicz-Sobolev spaces

We apply a three critical points theorem of B. Ricceri to establish the existence of at least three weak solutions for a class of non-homogeneous Neumann problems. Furthermore, by using another theorem of him, we prove that an appropriate oscillating behaviour of the nonlinear term ensures the existence of infinitely many weak solutions. Our analysis is based on recent variational methods for smooth functionals defined on Orlicz-Sobolev spaces.     

Applications of two critical point results for non-differentiable indefinite functionals

In this paper we study the existence of solution for two kinds of hemivariational inequalities: the first of them is of elliptic type, the second one of hamiltonian type. In those problems the energy functional is indefinite, so the classical variational principles can’t be used in a direct way. The results are an application of two theorems of existence of critical points for non-differentiable functionals recently obtained.     

Critical point theory for perturbations of symmetric functionals

Functionals which are invariant under the action of a compact transformation groupG often have many critical values. Here we consider functionals which are notG-invariant and give conditions for them to have infinitely many critical values; including a mountain pass theorem. We apply it to prove the existence of infinitely many solutions of a nonlinear Dirichlet problem with perturbedG-symmetries.     

Multiple periodic solutions for Hamiltonian systems with not coercive potential

Under an appropriate oscillating behavior of the nonlinear term, the existence of infinitely many periodic solutions for a class of second order Hamiltonian systems is established. Moreover, the existence of two non-trivial periodic solutions for Hamiltonian systems with not coercive potential is obtained, and the existence of three periodic solutions for Hamiltonian systems with coercive potential is pointed out. The approach is based on critical point theorems.     

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.     

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.     

Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems

We improve Benci and Rabinowitz's Linking theorem for strongly indefinite functionals, giving estimates for a suitably defined relative Morse index of critical points. Such abstract result is applied to the existence problem of periodic orbits and homoclinic solutions of first order Hamiltonian systems in cases where the Palais-Smale condition does not hold. Received January 27, 1999 / Accepted January 14, 2000 / Published online July 20, 2000     

On finding sign-changing solutions

Some parameter-depending linking theorems are established, which allow to produce a bounded and sign-changing Palais-Smale sequence. For even functionals, a parameter-depending fountain theorem is obtained which provides infinitely many bounded and sign-changing Palais-Smale sequences. A variant mountain pass theorem is built in cones which yields bounded, positive and negative Palais-Smale sequences. The usual Palais-Smale type compactness condition and its variants are completely not necessary for these theories. More exact locations of the critical sequences can be determined. The abstract results are applied to the Schrödinger equation with (or without) critical Sobolev exponents:

     

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．     

Minimax results for semicoercive functionals with application to differential equations

We give minimax theorems for some class of generalized convex and semicoercive functions. We define semicoercive saddle points and give sufficient conditions for functionals to have such critical points. Then we apply this method to show the existence of solutions for partial differential systems at resonance.      

An infinite dimensional Morse theory with applications

In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the one-dimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations.

     

Elliptic equations with indefinite concave nonlinearities near the origin

In this note existence of infinitely many solutions is proved for an elliptic equation with indefinite concave nonlinearities.     

A bifurcation result for semi-Riemannian trajectories of the Lorentz force equation

We obtain a bifurcation result for solutions of the Lorentz equation in a semi-Riemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the electromagnetic field. The flow of the Jacobi equation along each solution preserves the so-called electromagnetic symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution.We study electromagnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see (Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see (J. Funct. Anal. 162(1) (1999) 52)), we are able to prove that each non-degenerate and non-null electromagnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point.     

High energy solutions and nontrivial solutions for fourth-order elliptic equations

In this paper, we study the existence of nontrivial solutions and infinitely many high energy solutions for a class of nonlinear fourth-order elliptic equations in RN via variational methods. Three main theorems are obtained.