Multiple periodic solutions of Hamiltonian systems with prescribed energy

Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface Σ=H⁻¹(1). If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones.     

Homoclinic orbits for a nonperiodic Hamiltonian system

In this paper we prove the existence and multiplicity of homoclinic orbits for first order Hamiltonian systems of the form

     

Some existence results for a periodic problem with non-smooth potential

In this paper, two existence results for a class of second order periodic boundary value problems with non-smooth potential are obtained. We extend the Castro-Lazer-Thews reduction method to non-smooth functionals, the obtained result is then exploited to prove the existence of a nontrivial solution. Furthermore, we prove the existence of multiple solutions by using a multiplicity result based on local linking.     

Nontrivial periodic solutions for strong resonance Hamiltonian systems via a local linking theorem

In this paper, we first establish an existence result of critical points for a class of functionals defined on Hilbert spaces by using a local linking idea. Then as an application of the existence result, we obtain the existence of periodic solutions of strong resonance Hamiltonian systems which are asymptotically linear both at infinity and at origin.     

Large-time behavior of solutions to the equations of a viscous polytropic ideal gas

We study Lagrangian systems with symmetry under the action of a constant generalized force in the direction of the symmetry field. After Routh's reduction, such systems become nonautonomous with Lagrangian quadratic in time. We prove the existence of solutions tending to an orbit of the symmetry group as t± . As an example, we study doubly asymptotic solutions for the Kirchhoff problem of a heavy rigid body in an infinite volume of incompressible ideal fluid performing a potential motion.Supported by GNFM and by MURST (40%: «Equazioni di evoluzione...»).Supported by Russian Foundation of Basic Research and by INTAS.     

On the existence of H-surfaces into Riemannian manifolds

This paper considers the existence of a local minimizer of a conformally invariant functional defined on a space of maps of a closed Riemann surface into a compact Riemannian manifold . The functional is defined for a given tensor on of type (1,2) and we call its extremal an -surface. In fact, we prove that there exists a local minimizer of the functional in a given homotopy class under certain conditions on , and the minimum of the Dirichlet integral of maps of the homotopy class. Received January 21, 1994 / Received in revised form October 24, 1995 / Accepted March 15, 1996     

The existence of periodic solutions of non-autonomous second-order Hamiltonian systems

The purpose of this paper is to study the existence of periodic solutions for a class of non-autonomous second-order Hamiltonian systems. Some new existence theorems are obtained by using the least action principle and the saddle point theorem.     

A twist condition and periodic solutions of Hamiltonian systems

In this paper, we investigate existence of nontrivial periodic solutions to the Hamiltonian system

(HS)     

Multiple solutions for a class of nonlinear Schrödinger equations

In this paper, we find new conditions to ensure the existence of infinitely many homoclinic type solutions for the Schrödinger equation

     

A Nash type solution for hemivariational inequality systems

In this paper, we prove an existence result for a general class of hemivariational inequality systems using the Ky Fan version of the KKM theorem Fan (1984) [10] or Tarafdar fixed points Tarafdar (1987) [11]. As application, we give an infinite-dimensional version for the existence result of Nash generalized derivative points introduced recently by Kristály (2010) [5]. We also give an application to a general hemivariational inequality system.     

Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms

Based on new information concerning strongly indefinite functionals without Palais-Smale conditions, we study existence and multiplicity of solutions of the Schrödinger equation

     

Characterization of the Palais–Smale sequences for the conformal Dirac–Einstein problem and applications

In this paper we study the Palais–Smale sequences of the conformal Dirac–Einstein problem. After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the existence of infinitely many solutions to the problem provided that the underlying manifold exhibits certain symmetries.     

Existence of homoclinic solutions for a class of second-order Hamiltonian systems

A new result for existence of homoclinic orbits is obtained for the second-order Hamiltonian systems under a class of new superquadratic conditions. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of boundary-value problems which are obtained by the minimax methods.     

The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps

We prove a global existence of solutions for the Landau-Lifshitz equation of the ferromagnetic spin chain from am-dimensional manifoldM into the unit sphereS ² of ³ and establish some new links between harmonic maps and the solutions of the Landau-Lifshitz equation.     

Geodesics in stationary spacetimes and classical Lagrangian systems

We state a fundamental correspondence between geodesics on stationary spacetimes and the equations of classical particles on Riemannian manifolds, accelerated by a potential and a magnetic field. By variational methods, we prove some existence and multiplicity theorems for fixed energy solutions (joining two points or periodic) of the above described Riemannian equation. As a consequence, we obtain existence and multiplicity results for geodesics with fixed energy, connecting a point to a line or periodic trajectories, in (standard) stationary spacetimes.     

Cantor families of periodic solutions for wave equations via a variational principle

We prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a Cantor set of asymptotically full measure, via a variational principle. A Lyapunov-Schmidt decomposition reduces the problem to a finite dimensional bifurcation equation—variational in nature—defined on a Cantor set of non-resonant parameters. The Cantor gaps are due to “small divisors” phenomena. To solve the bifurcation equation we develop a suitable variational method. In particular, we do not require the typical “Arnold non-degeneracy condition” of the known theory on the nonlinear terms. As a consequence our existence results hold for new generic sets of nonlinearities.     

Bounded Palais-Smale sequences for functionals with a mountain pass geometry

We prove the existence of bounded Palais-Smale sequences for abstract functionals with a mountain pass geometry under hypotheses weaker than those commonly used in the literature. This is obtained via a generalization of a generic result of Jeanjean, combined with a rescaling argument. Applications to the existence of nontrivial solutions to semilinear elliptic problems are given. Received: 17 November 2005     

Singularly perturbed elliptic systems

We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations.     

Hamiltonian loops from the ergodic point of view

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S¹→G is called strictly ergodic if for some irrational number α the associated skew product map T:S¹×Y→S¹×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology. Received July 7, 1998 / final version received September 14, 1998