  This paper investigates the well-posedness of a type of state constraint
ergodic Mean Field Game system in a bounded domain in which the
Hamilton-Jacobi-Bellman equation is paired with an infinite Dirichlet boundary
condition. In this setting, the infinite boundary condition prevents the
underlying stochastic trajectories from reaching the boundary, resulting in a
state constraint solution. We consider a class of superlinear power-like
Hamiltonians and establish existence and uniqueness results under appropriate
assumptions on the coupling. We treat both non-local and local couplings, and
establish the uniqueness of the system by exploiting the decay of the density
near the boundary.
