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We consider the variational approach to prove the existence of solutions of second-order stationary Mean Field Games systems on a bounded domain $\Omega\subseteq {\mathbb R}^{d}$ with Neumann boundary conditions and with and without density constraints. We consider Hamiltonians which grow as $|\cdot|^{q'}$, where $q'=q/(q-1)$ and $q>d$. Despite this restriction, our approach allows us to prove the existence of solutions in the case of rather general coupling terms. When density constraints are taken into account, our results improve those in [A. R. Meszaros and F. J. Silva, J. Math. Pures Appl., 104 (2015), pp. 1135--1159]. Furthermore, our approach can be used to obtain solutions of systems with multiple populations.

On the Variational Formulation of Some Stationary Second-Order Mean Field Games Systems