Multiplicity of Solutions for a Class of Fourth-Order Elliptic Problems with Asymptotically Linear Term

Abstract

We study the following fourth-order elliptic equations: ${\Delta }^{2}u+a\Delta u=f\left(x,u\right),x\in \Omega ,u=\Delta u=0,\hspace{0.17em}x\in \partial \Omega$, where $\Omega \subset {ℝ}^N$ is a bounded domain with smooth boundary $\partial \Omega$ and $f(x,u)$ is asymptotically linear with respect to $u$ at infinity. Using an equivalent version of Cerami's condition and the symmetric mountain pass lemma, we obtain the existence of multiple solutions for the equations.