Infinitely many mountain pass solutions on a kind of fourth-order Neumann boundary value problem

Abstract

In this paper, the existence of infinitely many mountain pass solutions are obtained for the fourth-order boundary value problem (BVP) u ( 4 ) ( t ) - 2 u ″ ( t ) + u ( t ) = f ( u ( t ) ) , 0 < t < 1 , u ′ ( 0 ) = u ′ ( 1 ) = u ‴ ( 0 ) = u ‴ ( 1 ) = 0 , where f : R → R is continuous. The study of the problem is based on the variational methods and critical point theory. We prove the conclusion by using sub–sup solution method, Mountain Pass Theorem in Order Intervals, Leray–Schauder degree theory and Morse theory.