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.     

Infinitely many homoclinic solutions for second order Hamiltonian systems

In this paper we study the existence of infinitely many homoclinic solutions for second order Hamiltonian systems , , where L(t) is unnecessarily positive definite for all t∈R, and W(t,u) is of subquadratic growth as |u|→∞.     

Periodic solutions of a class of second order non-autonomous Hamiltonian systems

Some existence theorems are obtained for periodic solutions of non-autonomous second order systems by using the least action principle and minimax methods in critical point theory. Our results extend and improve many previously known results.     

Heteroclinic solutions for a class of the second order Hamiltonian systems

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system , where q∈Rn and V∈C¹(R×Rn,R), V?0. We will assume that V and a certain subset M⊂Rn satisfy the following conditions. M is a set of isolated points and #M?2. For every sufficiently small ε>0 there exists δ>0 such that for all (t,z)∈R×Rn, if d(z,M)?ε then −V(t,z)?δ. The integrals , z∈M, are equi-bounded and −V(t,z)→∞, as |t|→∞, uniformly on compact subsets of Rn?M. Our result states that each point in M is joined to another point in M by a solution of our system.     

Homoclinic solutions for a class of the second order Hamiltonian systems

We study the existence of homoclinic orbits for the second order Hamiltonian system , where q∈Rⁿ and V∈C¹(R×Rⁿ,R), V(t,q)=-K(t,q)+W(t,q) is T-periodic in t. A map K satisfies the “pinching” condition b₁|q|²?K(t,q)?b₂|q|², W is superlinear at the infinity and f is sufficiently small in L²(R,Rⁿ). A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second order differential equations.     

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

     

Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems

We prove the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems in of the form , where we assume the existence of a sequence such that and as for any . Moreover, under a suitable non degeneracy condition, we prove that this class of systems admits multibump solutions. Received February 2, 1996 / In revised form July 5, 1996 / Accepted October 10, 1996     

Homoclinic solutions for a class of second order Hamiltonian systems

In this paper, we study homoclinic solutions for the nonperiodic second order Hamiltonian systems where L is unnecessarily coercive or uniformly positively definite, and is only locally defined near the origin with respect to u. Under some general conditions on L and W, we show that the above system has infinitely many homoclinic solutions near the origin. Some related results in the literature are extended and generalized.     

Existence of homoclinic solutions to periodic orbits in a center manifold

Consider a Lagrangian of the form

     

Infinitely many homoclinic solutions for a second‐order Hamiltonian system

In this paper, we study the homoclinic solutions of the following second‐order Hamiltonian system where , and . Applying the symmetric Mountain Pass Theorem, we establish a couple of sufficient conditions on the existence of infinitely many homoclinic solutions. Our results significantly generalize and improve related ones in the literature. For example, is not necessary to be uniformly positive definite or coercive; through is still assumed to be superquadratic near , it is not assumed to be superquadratic near .     

Existence of homoclinic solutions for second order Hamiltonian systems with general potentials

In this paper we are concerned with the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian systems HS $$ \ddot{q}-L(t)q+W_{q}(t,q)=0, $$ where $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$ and $L\in C(\mathbb{R},\mathbb{R}^{n^{2}})$ is a symmetric and positive definite matrix for all $t\in\mathbb{R}$ . Assuming that the potential W satisfies some weaken global Ambrosetti-Rabinowitz conditions and L meets the coercive condition, we show that (HS) has at least one nontrivial homoclinic solution via using the Mountain Pass Theorem. Some recent results in the literature are generalized and significantly improved.     

Existence and concentration of positive solutions for a class of gradient systems

Some gradient systems with two competing potential functions are considered. Bound states (solutions with finite energy) are proved to exist and to concentrate at a point in the limit. The proof relies on variational methods, where the existence and concentration of positive solutions are related to a suitable ground energy function.     

Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system

This paper deals with existence and exponential decay of homoclinic orbits in the first-order Hamiltonian system

     

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     

Multibump homoclinic solutions for a class of second order, almost periodic Hamiltonian systems

We prove the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems in R ^N of the form where is almost periodic and W is superquadratic. Received October 17, 1995     

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

     

A multiplicity result for a class of strongly indefinite asymptotically linear second order systems

We prove a multiplicity result for a class of strongly indefinite nonlinear second order asymptotically linear systems with Dirichlet boundary conditions. The key idea for the proof is to bring together the classical shooting method and the Maslov index of the linear Hamiltonian systems associated to the asymptotic limits of the given nonlinearity.