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Infinitely many homoclinic solutions for a second-order Hamiltonian system with locally defined potentials

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  • Wang, Xiaoping

Abstract

In this paper, we study homoclinic solutions of the following second-order Hamiltonian system u¨(t)−L(t)u(t)+∇W(t,u(t))=0,where t∈R,u∈RN,L:R→RN×N and W:R×RN→R. Applying a new symmetric Mountain Pass Theorem established by Kajikiya, we prove the existence of infinitely many homoclinic solutions for the above system in the case where L(t) is coercive but unnecessarily positive definite for all t∈R, and W(t, x) is only locally defined near the origin with respect to x. Our results significantly generalize and improve related ones in the literature.

Suggested Citation

  • Wang, Xiaoping, 2016. "Infinitely many homoclinic solutions for a second-order Hamiltonian system with locally defined potentials," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 47-50.
    • Handle: RePEc:eee:chsofr:v:87:y:2016:i:c:p:47-50
    DOI: 10.1016/j.chaos.2016.02.034
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    References listed on IDEAS

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    1. Sun, Juntao & Wu, Tsung-fang, 2015. "Homoclinic solutions for a second-order Hamiltonian system with a positive semi-definite matrix," Chaos, Solitons & Fractals, Elsevier, vol. 76(C), pages 24-31.
    2. Ying Lv & Chun-Lei Tang, 2013. "Existence and Multiplicity of Homoclinic Orbits for Second-Order Hamiltonian Systems with Superquadratic Potential," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, February.
    3. X. H. Tang, 2016. "Infinitely many homoclinic solutions for a second-order Hamiltonian system," Mathematische Nachrichten, Wiley Blackwell, vol. 289(1), pages 116-127, January.
    4. Lv, Ying & Tang, Chun-Lei, 2013. "Homoclinic orbits for second-order Hamiltonian systems with subquadratic potentials," Chaos, Solitons & Fractals, Elsevier, vol. 57(C), pages 137-145.
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    1. Sun, Juntao & Wu, Tsung-fang, 2015. "Homoclinic solutions for a second-order Hamiltonian system with a positive semi-definite matrix," Chaos, Solitons & Fractals, Elsevier, vol. 76(C), pages 24-31.

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