Hits:

Indexed by:期刊论文

Date of Publication:2015-01-19

Journal:ELECTRONIC JOURNAL OF DIFFERENTIAL EQUATIONS

Included Journals:SCIE

ISSN No.:1072-6691

Key Words:Homoclinic solution; Mountain pass theorem; damped differential equation; Nehari manifold

Abstract:We study the existence and multiplicity of homoclinic solutions for the second-order damped differential equation
   u + c(u)overdot - 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 in finitely 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.