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.

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.