Multiple solutions to fourth-order boundary value problems

AbstractIn 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)) for all t∈[0,1] subject to Dirichlet boundary value condition, where f∈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 f which are able to guarantee that this boundary value problem has at least one nontrivial, two nontrivial, m distinct pairs of solutions, and infinitely many solutions, respectively. Our results improve some recent works