Nonlocal coupled systems driven by fourth-order Leray–Lions operators in variable exponent spaces

This paper studies the existence of solutions for a class of coupled systems governed by a fourth-order Leray–Lions operator with nonlinear boundary conditions and nonlocal terms. The domain is assumed to be a bounded subset of RN with N>2, and the boundary is of class C1. The system involves a set of Carathéodory-type functions satisfying specific growth conditions and parameter assumptions. The nonlocality arises from integral terms over the domain, preventing the equations from holding as pointwise identities. We establish the existence of at least one solution and, under certain conditions, prove the existence of multiple solutions. In particular, we show that the system admits at least three distinct solutions for appropriate parameter values. The results are derived using variational methods and critical point theory, providing a comprehensive analysis of the solution structure for the problem.