Remarks on some fourth-order boundary value problems with non-monotone increasing eigenvalue sequences

Abstract

By $Z_{2}$-index theory, the existence and multiplicity of solutions for some fourth-order boundary value problems $$ \begin{cases} u^{(4)}+au''=\mu u+F_{u}(t,u),\ 00$ and $\mu\in \mathbb{R}$ is an eigenvalue of the corresponding eigenvalue problem. The difficulty caused by the non-monotone eigenvalue sequence is handled concretely.

Keywords

• fourth-order differential equation
• boundary value problem
• resonance
• critical point
• $z_{2}$-index theory