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Homoclinic solutions for second order Hamiltonian systems with general potentials

  • Ziheng Zhang EMAIL logo , Honglian You and Rong Yuan
Published/Copyright: November 3, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 4

Abstract

In this paper we are concerned with the existence of infinitely many homoclinic solutions for the following second order non-autonomous Hamiltonian systems

u¨t−Ltut+∇Wt,ut=0(HS)

where t ∈ ℝ, L ∈ C(ℝ, ℝn2) is a symmetric and positive definite matrix for all t ∈ ℝ, W ∈ C1(ℝ × ℝn, ℝ) and ∇W(t,u) is the gradient of W at u. The novelty of this paper is that, assuming that L meets some coercive condition and the potential W is of the form W(t, u) = W1(t, u) + W2(t, u), for the first time we show that (HS) possesses two different sequences of infinitely many homoclinic solutions via the Fountain theorem and the dual Fountain theorem such that the corresponding energy functional of (HS) goes to infinity and zero, respectively. Some recent results in the literature are generalized and significantly improved.

MSC 2010: Primary 34C37; 35A15; 35B38
Keywords: homoclinic solutions; critical point; variational methods; fountain theorem

(Communicated by Michal Feckan)

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Received: 2013-9-9
Revised: 2014-2-12
Published Online: 2016-11-3
Published in Print: 2016-8-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2015-0190
Keywords for this article
homoclinic solutions; critical point; variational methods; fountain theorem

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