Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 56, pp. 1-10.

Title: Positive solutions for n-dimensional fourth-order systems under a parametric condition

Authors: Pablo Alvarez-Caudevilla (Univ. Carlos III de Madrid, Spain)
         Cristina Brandle  (Univ. Carlos III de Madrid, Spain)
         Devashish Sonowal (Univ. Carlos III de Madrid, Spain)

Abstract:
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.

Submitted August 14, 2024. Published May 29, 2025.
Math Subject Classifications: 35J70, 35J47, 35K57.
Key Words: Coupled system; higher order operator.