  We investigate the existence of classical solutions to second-order quadratic
Mean-Field Games systems with local and strongly decreasing couplings of the
form $-\sigma m^\alpha$, $\alpha \ge 2/N$, where $m$ is the population density
and $N$ is the dimension of the state space. We prove the existence of
solutions under the assumption that $\sigma$ is small enough. For large
$\sigma$, we show that existence may fail whenever the time horizon $T$ is
large.

Analysis of PDEs