Anisotropic Kirchhoff systems with singular terms and critical exponential growth: existence and numerical evidence via the Finsler–Laplacian

This paper investigates the existence, positivity, and qualitative behavior of weak solutions for a class of generalized Kirchhoff-type elliptic systems driven by the anisotropic Finsler–Laplacian operator. The system involves a nonlocal Kirchhoff coefficient depending on the anisotropic Dirichlet energy, combined with singular nonlinearities and source terms exhibiting subcritical, critical (in the sense of the Trudinger–Moser threshold) exponential growth of Trudinger–Moser type. Owing to the anisotropic structure of the operator and the presence of exponential nonlinearities, classical Sobolev embedding theorems are no longer applicable, which significantly increases the analytical difficulty of the problem. By employing variational techniques, Galerkin approximations, and a sharp anisotropic Trudinger–Moser inequality, we establish the existence of at least one positive weak solution under suitable assumptions on the singular and nonlocal terms. The analysis extends several classical results for Kirchhoff-type problems involving the standard Laplacian to a fully anisotropic Finsler–Laplacian framework within a rigorous subcritical and critical setting. In addition, a comprehensive numerical study based on finite element discretization and iterative schemes is carried out to illustrate and validate the theoretical findings. The numerical results confirm the existence, positivity, and stability of solutions across subcritical, critical (and numerically explored supercritical) regimes, and they highlight the regularizing role of the nonlocal Kirchhoff function as well as the influence of the singular parameters. These computations provide strong quantitative support for the analytical results and demonstrate the robustness of the proposed variational framework.