Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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INFINITELY MANY HOMOCLINIC SOLUTIONS FOR DAMPED VIBRATION SYSTEMS WITH LOCALLY DEFINED POTENTIALS

  • Received : 2021.01.07
  • Accepted : 2021.11.09
  • Published : 2022.07.31
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Abstract

In this paper, we are concerned with the existence of infinitely many fast homoclinic solutions for the following damped vibration system $$(1){\hspace{32}}{\ddot{u}}(t)+q(t){\dot{u}}(t)-L(t)u(t)+{\nabla}W(t,u(t))=0,\;{\forall}t{\in}{\mathbb{R}},$$ where q ∈ C(ℝ, ℝ), L ∈ C(ℝ, ${\mathbb{R}}^{N^2}$) is a symmetric and positive definite matix-valued function and W ∈ C1(ℝ×ℝN, ℝ). The novelty of this paper is that, assuming that L is bounded from below unnecessarily coercive at infinity, and W is only locally defined near the origin with respect to the second variable, we show that (1) possesses infinitely many homoclinic solutions via a variant symmetric mountain pass theorem.

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Acknowledgement

The authors thank the referee for valuable comments and suggestions that improved the paper.

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