Multi-population Mean Field Games systems with Neumann boundary conditions

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Stationary focusing mean-field games (2016, background): “Existence of smooth solutions for (10) is still an open problem for general power-like nonlinearities, and known results require conditions on the exponents α, γ; see [14, 18, 19, 34].”
Mean field games models of segregation (2016, background): “[20]), provided the negative part (·)− inG is replaced by some smooth regularization.”
Nonlinear elliptic systems and mean-field games (2016, methodology): “For some very special cases of uniqueness with two populations see Cirant [25, 26] (where Neumann boundary conditions are also treated).”
On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games (2017, unknown): “In this case the monotonicity conditions are more restrictive: in addition to a form of aversion to crowd in each population, they require that the costs of intraspeciﬁc interactions are larger than the costs of the interactions between the two populations ( cf. [ 15]).”
Regularity for second order stationary mean-field games (2015, methodology): “Recent developments on stationary mean-field game problems were reported in [7], where the integral Bernstein estimates were used in the context of mean-field games for the first time (see also [21]).”