  Mean-field games (MFGs) are models for large populations of competing
rational agents that seek to optimize a suitable functional. In the case of
congestion, this functional takes into account the difficulty of moving in
high-density areas. Here, we study stationary MFGs with congestion with
quadratic or power-like Hamiltonians. First, using explicit examples, we
illustrate two main difficulties: the lack of classical solutions and the
existence of areas with vanishing density. Our main contribution is a new
variational formulation for MFGs with congestion. This formulation was not
previously known, and, thanks to it, we prove the existence and uniqueness of
solutions. Finally, we consider applications to numerical methods.
