Electron. J. Diff. Equ.,
Vol. 2015 (2015), No. 15, pp. 117.
Homoclinic and quasihomoclinic solutions for damped differential equations
ChuanFang Zhang, ZhiQing Han
Abstract:
We study the existence and multiplicity of homoclinic solutions for the
secondorder damped differential equation
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 subquadratic or superquadratic
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 quasihomoclinic 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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ChuanFang Zhang
School of Mathematical Sciences
Dalian University of Technology
116024 Dalian, China
email: kyzcf2006@163.com


ZhiQing Han
School of Mathematical Sciences
Dalian University of Technology
116024 Dalian, China
email: hanzhiq@dlut.edu.cn

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