Abstract: In this paper, we investigate the Dirichlet eigenvalue problem of fourth-order elliptic system of differential equations on an n-dimensional Euclidean space as follows
\begin{equation*}
\left\{\begin{aligned}
&A\Delta^2\boldsymbol{u} = -\Gamma\Delta \boldsymbol{u}, && \text{in} \quad \Omega, \\
&\boldsymbol{u} = \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{\nu}} = \boldsymbol{0}, && \text{on} \quad \partial \Omega,
\end{aligned}\right.
\end{equation*}
where A is a symmetric coefficient matrix and ν is the outward unit normal vector field of ∂Ω. We derive some inequalities for lower order eigenvalues of this problem. Our results cover some previous results for the buckling problem.

Key words: Eigenvalue, Inequality, Elliptic system of differential equations