Multiple solutions to fourth-order boundary value problems

Abstract

In this paper, we study the existence and multiplicity of solutions to the fourth-order boundary value problem $u^{\left(4\right)}\left(t\right)+\beta u^{″}\left(t\right)-\alpha u\left(t\right)=f(t,u\left(t\right))$ for all $t\in [0,1]$ subject to Dirichlet boundary value condition, where $f\in C^1([0,1]\times {\mathbb{R}}^1,{\mathbb{R}}^1),\alpha ,\beta \in {\mathbb{R}}^1$. 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.