Two homoclinic solutions for a nonperiodic fourth‐order differential equation without coercive condition

In this paper, we investigate the existence of homoclinic solutions for a class of fourth‐order nonautonomous differential equationsu ( 4 ) + w u ′ ′ + a ( x ) u = f ( x , u ) ,where w is a constant, a ∈ C ( R , R ) and f ∈ C ( R × R , R ) . By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that a ( x ) and f ( x , u ) are nonperiodic in x , a does not fulfil the coercive condition, and f does not satisfy the well‐known ( A R )‐condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth‐order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.