Nontrivial solutions for nonlinear discrete boundary value problems of the fourth order

Abstract

We study the existence of multiple nontrivial solutions for nonlinear fourth order discrete boundary value problems. We first establish criteria for the existence of at least two nontrivial solutions of the problems and obtain conditions to guarantee that the two solutions are sign-changing. Under some appropriate assumptions, we further prove that the problems have at least three nontrivial solutions, which are positive, negative, and sign-changing, respectively. We include two examples to illustrate the applicability of our results. Our theorems are proved by employing variational approaches, combined with the classic mountain pass lemma and a result from the theory of invariant sets of descending flow.