Existence of weak solutions to stationary mean-field games through variational inequalities

Here, we consider stationary monotone mean-field games (MFGs) and study the existence of weak solutions induced by monotonicity. First, we introduce a regularized problem that preserves the monotonicity. Next, using variational inequality techniques, we prove the existence of solutions to the regularized problem. Then, using Minty's method, we establish the existence of weak solutions for the original MFG. Finally, we examine the properties of these weak solutions in several examples. Our methods provide a general framework to construct weak solutions to stationary MFGs with local, nonlocal, or congestion terms.