On some singular mean-field games

Here, we prove the existence of smooth solutions for mean-field games with a singular mean-field coupling; that is, a coupling in the Hamilton-Jacobi equation of the form \begin{document}$ g(m) = -m^{- \alpha} $\end{document} with \begin{document}$ \alpha>0 $\end{document} . We consider stationary and time-dependent settings. The function \begin{document}$ g $\end{document} is monotone, but it is not bounded from below. With the exception of the logarithmic coupling, this is the first time that MFGs whose coupling is not bounded from below is examined in the literature. This coupling arises in models where agents have a strong preference for low-density regions. Paradoxically, this causes the agents move towards low-density regions and, thus, prevents the creation of those regions. To prove the existence of solutions, we consider an approximate problem for which the existence of smooth solutions is known. Then, we prove new a priori bounds for the solutions that show that \begin{document}$ \frac 1 m $\end{document} is bounded. Finally, using a limiting argument, we obtain the existence of solutions. The proof in the stationary case relies on a blow-up argument and in the time-dependent case on new bounds for \begin{document}$ m^{-1} $\end{document} .