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New Homoclinic Orbits for Hamiltonian Systems with Asymptotically Quadratic Growth at Infinity

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Abstract

In this paper, we study the existence and multiplicity of homoclinic solutions for following Hamiltonian systems with asymptotically quadratic nonlinearities at infinity

$$\begin{aligned} \ddot{u}(t)-L(t)u+\nabla W(t,u)=0. \end{aligned}$$

We introduce a new coercive condition and obtain a new embedding theorem. With this theorem, we show that above systems possess at least one nontrivial homoclinic orbits by generalized mountain pass theorem. By variant fountain theorem, infinitely many homoclinic orbits are obtained for above problem with symmetric condition. Our asymptotically quadratic conditions are different from previous ones in the references.

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Authors and Affiliations

  1. College of Science, Southwest Petroleum University, Chengdu, 610500, Sichuan, People’s Republic of China

    Dong-Lun Wu

  2. Institute of Nonlinear Dynamics, Southwest Petroleum University, Chengdu, 610500, Sichuan, People’s Republic of China

    Dong-Lun Wu

  3. School of Economic and Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, Sichuan, People’s Republic of China

    Xiang Yu

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  1. Dong-Lun Wu
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  2. Xiang Yu
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Correspondence to Dong-Lun Wu.

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This werk is partially supported by NSF of China (No.11801472) and NSF of China (No.11701464) and the Youth Science and Technology Innovation Team of Southwest Petroleum University for Nonlinear Systems (No.2017CXTD02) and the Fundamental Research Funds for the Central Universities, P.R. China (JBK1805001).

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Wu, DL., Yu, X. New Homoclinic Orbits for Hamiltonian Systems with Asymptotically Quadratic Growth at Infinity. Qual. Theory Dyn. Syst. 19, 22 (2020). https://doi.org/10.1007/s12346-020-00346-9

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  • DOI: https://doi.org/10.1007/s12346-020-00346-9

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