On an n-dimensional fourth-order system under a parametric condition

We establish the existence of positive solutions for a system of coupled fourth-order partial differential equations on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$\begin{align*} \left{\begin{array}{l} Δ^2u_1 +β_1 Δu_1-α_1 u_1=f_1({ x},u_1,u_2),\Δ^2 u_2+β_2Δu_2-α_2 u_2=f_2({ x},u_1,u_2), \end{array} \quad \quad x\inΩ, \right. \end{align*}subject to homogeneous Navier boundary conditions, where the functions $f_1,f_2:\mathrm{\Omega }\times [0,\mathrm{\infty })\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })$ 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.