Positive solutions for n-dimensional fourth-order systems 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 \(\Omega \subset \mathbb{R}^n\), $$\displaylines{ \Delta^2u_1 +\beta_1 \Delta u_1-\alpha_1 u_1=f_1({ x},u_1,u_2),\cr \Delta^2 u_2+\beta_2\Delta u_2-\alpha_2 u_2=f_2({ x},u_1,u_2), }$$ for \(x\in\Omega\), subject to homogeneous Navier boundary conditions, where the functions \(f_1,f_2 : \Omega\times [0,\infty)\times [0,\infty) \to [0,\infty)\) 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. For more information see https://ejde.math.txstate.edu/Volumes/2025/56/abstr.html