Superquadratic Hamiltonian Systems Research Articles

On locally superquadratic Hamiltonian systems with periodic potential

In this paper, we study the second-order Hamiltonian systems u¨−L(t)u+∇W(t,u)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\ddot{u}-L(t)u+\\nabla W(t,u)=0, $$\\end{document} where tin mathbb{R}, uin mathbb{R}^{N}, L and W depend periodically on t, 0 lies in a spectral gap of the operator -d^{2}/dt^{2}+L(t) and W(t,x) is locally superquadratic. Replacing the common superquadratic condition that lim_{|x|rightarrow infty }frac{W(t,x)}{|x|^{2}}=+infty uniformly in tin mathbb{R} by the local condition that lim_{|x|rightarrow infty }frac{W(t,x)}{|x|^{2}}=+infty a.e. tin J for some open interval Jsubset mathbb{R}, we prove the existence of one nontrivial homoclinic soluiton for the above problem.

EXISTENCE OF PERIODIC SOLUTIONS FOR HAMILTONIAN SYSTEMS WITH SUPER-LINEAR AND SIGN-CHANGING NONLINEARITIES

In this paper, we consider the existence of periodic solutions for the super quadratic second order Hamiltonian system, and primitive functions of nonlinearities are allowed to be sign-changing. By using some weaker conditions, our result extends and improves some existed results in the literature.

Subharmonic Pl-solutions of first order Hamiltonian systems

In this paper, for any symplectic matrix P, the existence of subharmonic Pl-solutions of the first order non-autonomous superquadratic Hamiltonian systems is considered. Under the convex condition, the existence of infinitely many geometrically distinct Pl-solutions is proved.

One Generalized Critical Point Theorem and its Applications on Super-quadratic Hamiltonian Systems

In this paper, we prove a generalized critical point theorem under the condition (C), which is weaker than the (PS) condition. As its applications, we obtain the existence of the solutions for the Hamiltonian systems with a new super-quadratic conditions generalizing one in papers [2] and [12].

Multiplicity of subharmonic solutions and periodic solutions of a particular type of super-quadratic Hamiltonian systems

This paper considers the Hamiltonian systemswith new generalized super-quadratic conditions. Using the variational principle and critical point theory,we obtain infinitely many distinct subharmonic solutions and an unbounded sequence of periodic solutions.

Existence of periodic solutions for a class of second-order discrete Hamiltonian systems

By using the variant fountain theorem, we study the existence of periodic solutions for a class of superquadratic non-autonomous second-order discrete Hamiltonian systems.

On differential systems with strongly indefinite variational structure

We obtain multiplicity results for a class of first-order superquadratic Hamiltonian systems and a class of indefinite superquadratic elliptic systems which lead to the study of strongly indefinite functionals. There is no assumption to the effect that the nonlinear terms have to satisfy the Ambrosetti–Rabinowitz superquadratic condition. To establish the existence of solutions, a new version of the symmetric mountain pass theorem for strongly indefinite functionals is presented in this paper. This theorem is subsequently applied to deal with cases where all the Palais–Smale sequences of the energy functional may be unbounded.

Existence of periodic solutions of a particular type of super-quadratic Hamiltonian systems

This paper considers the Hamiltonian systems with new super-quadratic conditions covering the case Hˆ(t,z)=(|p|1+μν+|q|1+νμ)(ln⁡(1+|z|ξˆ))ηˆ, where ξˆ,ηˆ,μ,ν>1 with 1μ+1ν<1 and μ2ν<νμ<μν<1+νμ. Using the mini-max principle, we obtain the existence of the periodic solutions.

Existence of Periodic Solution for a Class of Symmetric Superquadratic Second-Order Non-Autonomous Hamiltonian Systems

In the paper, by the symmetrical Mountain-Pass lemma in critical point theory, the existence of infinitely anti-and odd periodic solutions with a fixed period is obtained for a class of symmetric superquadratic non-autonomous Hamiltonian systems.

Ground-state solutions for superquadratic Hamiltonian elliptic systems with gradient terms

In this paper, we study the following Hamiltonian elliptic system with gradient terms: {−Δu+b(x)⋅∇u+V(x)u=Hv(x,u,v),−Δv−b(x)⋅∇v+V(x)v=Hu(x,u,v) for x∈RN, where V(x), b(x), and H(x,u,v) are 1-periodic in x. Under weak superquadratic assumptions for the nonlinearities, we use much more direct methods to prove that all the Cerami sequences for the energy functional are bounded and establish the existence of ground-state solutions for this system.

Solutions of nonperiodic super quadratic Hamiltonian systems

This paper concerns solutions for the Hamiltonian system: , where H(t,u) = 1 ∕ 2Lu ⋅ u + W(t,u), L is a 2N × 2N symmetric matrix, and . We consider the case that 0 ∉ σc( − (Jd ∕ dt + L)) and W satisfies some new generalized super quadratic condition different from the type of Ambrosetti–Rabinowitz. The method is variational: By virtue of some auxiliary system related to the ‘limit equation’ of the Hamiltonian system, we first establish that (C)c‐condition holds true for all c less than the least energy of the limit equation. Then, using some weak linking theorem recently developed, we obtain one least energy solution of the Hamiltonian system. Copyright © 2013 John Wiley &amp; Sons, Ltd.

On the existence of periodic and homoclinic orbits for first order superquadratic Hamiltonian systems

In this paper we consider the following Hamiltonian system $$J\dot u + B(t)u +\nabla W(t,u)=0.\quad\quad (HS)$$ Under a new superquadratic assumption on the potential, we prove that (HS) has a sequence of subharmonics. This will be done using a minimax result in critical point theory. Also, we study the asymptotic behavior of these subharmonics and we establish the existence of a homoclinic orbit for (HS). Previous results in the topic, mainly those due to Rabinowitz and Tanaka, are significantly improved.

Existence of infinitely many homoclinic orbits for nonperiodic superquadratic Hamiltonian systems

In this paper, we study the following nonperiodic Hamiltonian system ż=JHz(t,z), where H(t,z) is superquadratic in z∈R2N as |z|→∞. By applying a generalized linking theorem for strongly indefinite functionals, we obtain infinitely many homoclinic orbits for the above system.

Erratum to: Homoclinic orbits for an unbounded superquadratic Hamiltonian systems

Erratum to: Homoclinic orbits for an unbounded superquadratic Hamiltonian systems

The existence of solutions for superquadratic Hamiltonian elliptic systems on [formula omitted

This paper is concerned with the following Hamiltonian elliptic system: { − △ u + V ( x ) u = H u ( x , u , v ) in R N ; − △ v + V ( x ) v = − H v ( x , u , v ) in R N ; u ( x ) → 0 and v ( x ) → 0 as | x | → ∞ , where V ( x ) ∈ C ( R N , R ) and H ( x , z ) is superquadratic in z as | z | → ∞ with z : = ( u , v ) . By applying a critical point theorem for strongly indefinite functionals, we obtain infinitely many solutions for the above system.

Existence of solutions for nonperiodic superquadratic Hamiltonian elliptic systems

This paper is concerned with the following nonperiodic Hamiltonian elliptic system { − Δ u + V ( x ) u = R v ( x , u , v ) in R N , − Δ v + V ( x ) v = R u ( x , u , v ) in R N , u ( x ) → 0 and v ( x ) → 0 as | x | → ∞ , where R ( x , z ) is superquadratic in z as | z | → ∞ with z = ( u , v ) . By applying a critical point theorem for strongly indefinite functionals, we prove the existence of solution for the above system.

Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems

In this paper, we find new conditions, which are different from those used in previous related studies, to ensure the existence of infinitely many homoclinic orbits for the second order Hamiltonian systems of the form − q ̈ = V q ( t , q ) . Here, we assume that V ( t , q ) depends periodically on t , and assume, on q , that V ( t , q ) is asymptotically quadratic at q = 0 and is, as | q | → ∞ , either asymptotically quadratic or superquadratic, as well as the new conditions.

Nontrivial solutions of superquadratic Hamiltonian systems with Lagrangian boundary conditions and the L-index theory

In this paper, the authors study the existence of nontrivial solutions for the Hamiltonian systems \( \dot z \)(t) = J▽H(t, z(t)) with Lagrangian boundary conditions, where Open image in new window is a semipositive symmetric continuous matrix and Open image in new window satisfies a superquadratic condition at infinity. We also obtain a result about the L-index.

Periodic solutions for a class of superquadratic Hamiltonian systems

We prove the existence of nontrivial critical points for a class of superquadratic nonautonomous second-order Hamiltonian systems by applying condition ( C ) ∗ to critical point theory, and some new solvability conditions of nontrivial periodic solutions are obtained.

Periodic solutions for a class of first order super-quadratic Hamiltonian system

Based on a developed local linking theorem, by replacing the Palais–Smale condition with a Cerami one, we consider the following unbounded Hamiltonian system: (0.1) { ∂ t u − Δ x u + V ( x ) u = H v ( t , x , u , v ) , − ∂ t v − Δ x v + V ( x ) v = H u ( t , x , u , v ) for ( t , x ) ∈ R × Ω , where Ω ⊂ R N , N ⩾ 1 , is a bounded domain with smooth boundary ∂Ω. The nonlinearities are assumed to be super-quadratic at the infinite, which are different from those investigated previously.