  Here, we study the existence and the convergence of solutions for the
vanishing discount MFG problem with a quadratic Hamiltonian. We give conditions
under which the discounted problem has a unique classical solution and prove
convergence of the vanishing-discount limit to a unique solution up to
constants. Then, we establish refined asymptotics for the limit. When those
conditions do not hold, the limit problem may not have a unique solution and
its solutions may not be smooth, as we illustrate in an elementary example.
Finally, we investigate the stability of regular weak solutions and address the
selection problem. Using ideas from Aubry-Mather theory, we establish a
selection criterion for the limit.
