A system of coupled anisotropic operators with gradient-dependent nonlinearities

In this paper, we study a system of two coupled anisotropic elliptic equations involving gradient-dependent nonlinearities and unbounded coefficients. Each equation is governed by a generalized p-Laplacian operator with a coefficient that may depend on the solution itself. The coupling appears both in the source terms and through the gradient of the unknowns. By employing the sub-supersolution method in combination with truncation techniques and Moser iteration, we prove the existence of weak solutions lying within a prescribed ordered interval. Furthermore, we show that all weak solutions are uniformly bounded in the appropriate Sobolev space. The results are applicable to strongly coupled reaction-diffusion processes and multi-component systems where the flux of one species is influenced by the gradient of another.