Homoclinic solutions for a nonperiodic fourth order differential equations without coercive conditions

In this paper we investigate the existence of homoclinic solutions for the following fourth order nonautonomous differential equations u(4)+wu″+a(x)u=f(x,u),(FDE) wherew is a constant, a∈C(R,R) and f∈C(R×R,R). The novelty of this paper is that, when (FDE) is nonperiodic, i.e., a and f are nonperiodic in x and assuming that a does not fulfil the coercive conditions and f satisfies some more general (AR) condition, we establish one new criterion to guarantee that (FDE) has at least one nontrivial homoclinic solution via using the Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved.