Electron. J. Diff. Equ., Vol. 2015 (2015), No. 15, pp. 1-17.

Homoclinic and quasi-homoclinic solutions for damped differential equations

Chuan-Fang Zhang, Zhi-Qing Han

Abstract:
We study the existence and multiplicity of homoclinic solutions for the second-order damped differential equation
$$ \ddot{u}+c\dot{u}-L(t)u+W_u(t,u)=0, $$
where L(t) and W(t,u) are neither autonomous nor periodic in t. Under certain assumptions on L and W, we obtain infinitely many homoclinic solutions when the nonlinearity W(t,u) is sub-quadratic or super-quadratic by using critical point theorems. Some recent results in the literature are generalized, and the open problem proposed by Zhang and Yuan is solved. In addition, with the help of the Nehari manifold, we consider the case where W(t,u) is indefinite and prove the existence of at least one nontrivial quasi-homoclinic solution.

Submitted August 19, 2014. Published January 19, 2015.
Math Subject Classifications: 34C37, 35A15, 37J45.
Key Words: Homoclinic solution; Mountain pass theorem; damped differential equation; Nehari manifold.

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Chuan-Fang Zhang
School of Mathematical Sciences
Dalian University of Technology
116024 Dalian, China
email: kyzcf2006@163.com
Zhi-Qing Han
School of Mathematical Sciences
Dalian University of Technology
116024 Dalian, China
email: hanzhiq@dlut.edu.cn

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