An elliptic partial differential equations system and its application involving the Biharmonic operator

Abstract

This paper investigates a novel system of coupled fourth-order nonlinear partial differential equations defined on the whole space R^N. The system is characterized by a unique combination of the biharmonic operator, the Laplacian, and nonlinear terms involving the square of the gradient. The coupling is linear, involving positive parameters lambda_i, k_i, and a_i, alongside continuous source functions f_i. We establish the existence of solutions using the sub- and super-solution method. Additionally, we discuss an engineering application to illustrate the physical relevance of the problem. To the best of our knowledge, this is the first study addressing a system that integrates this specific configuration of biharmonic, Laplacian, and gradient terms.