$$C^{1,\alpha }$$ regularity for stationary mean-field games with logarithmic coupling

This paper investigates stationary mean-field games (MFGs) on the torus with Lipschitz non-homogeneous diffusion and logarithmic-like couplings. The primary objective is to understand the existence of \(C^{1,\alpha }\) solutions to address the research gap between low-regularity results for bounded and measurable diffusions and the smooth results modeled by the Laplacian. We use the Hopf-Cole transformation to convert the MFG system into a scalar elliptic equation. Then, we apply Morrey space methods to establish the existence and regularity of solutions. The introduction of Morrey space methods offers a novel approach to address regularity issues in the context of MFGs.