  We establish the existence of positive solutions for a system of coupled
fourth-order partial differential equations on a bounded domain $\Omega \subset
\mathbb{R}^n$\begin{align*}
  \left\{\begin{array}{l} \Delta^2u_1 +\beta_1 \Delta u_1-\alpha_1 u_1=f_1({
x},u_1,u_2),\\\Delta^2 u_2+\beta_2\Delta u_2-\alpha_2 u_2=f_2({ x},u_1,u_2),
\end{array} \quad \quad x\in\Omega,
  \right. \end{align*}subject to homogeneous Navier boundary conditions, where
the functions $f_1,f_2 : \Omega\times [0,\infty)\times [0,\infty) \rightarrow
[0,\infty)$ are continuous, and $\alpha_1,\alpha_2,\beta_1$ and $\beta_2$ are
real parameters satisfying certain constraints related to the eigenvalues of
the associated Laplace operator.
