Multiple Solutions for a Nonlocal Anisotropic Schrödinger–Kirchhoff System with Applications to Mining Excavation Stability

This paper studies the existence of multiple weak solutions for a Schrödinger–Kirchhoff-type elliptic system driven by nonlocal anisotropic variable exponent operators. The model combines anisotropic μ(·)-Laplacian-type diffusion, Kirchhoff nonlocal terms depending on global energy integrals, and nonlinear potential effects on RN. The analysis is carried out in anisotropic variable exponent Sobolev spaces, and the main result is obtained via variational methods based on Ricceri’s three critical points theorem. As an application, the theoretical framework is used to investigate the stability of underground excavations in geologically heterogeneous mining formations, where the anisotropic exponents capture direction-dependent rock resistance and the Kirchhoff terms model strain-dependent stiffness; a numerical example illustrates the existence of multiple equilibrium states and identifies a critical loading threshold separating single- and multiple-equilibrium regimes.