Multiple solutions to fourth-order boundary value problems

In this paper, we study the existence and multiplicity of solutions to the fourth-order boundary value problem u(4)(t)+βu″(t)−αu(t)=f(t,u(t))u(4)(t)+βu″(t)−αu(t)=f(t,u(t)) for all t∈[0,1]t∈[0,1] subject to Dirichlet boundary value condition, where f∈C1([0,1]×R1,R1),α,β∈R1f∈C1([0,1]×R1,R1),α,β∈R1. By using the critical point theory and the infinite dimensional Morse theory, we establish some conditions on ff which are able to guarantee that this boundary value problem has at least one nontrivial, two nontrivial, mm distinct pairs of solutions, and infinitely many solutions, respectively. Our results improve some recent works.