Existence of solutions and selection problem for quasi-stationary contact mean field games

First, we study the existence of solutions for a class of first order mean field games systems H ( x , u , D u ) = F ( x , m ( t )), x ∈ M , ∀ t ∈ [0, T ], ∂ t m - div (m ∂H / ∂ p ( x, u, Du ) = 0, ( x, t ) ∈ M × (0, T ], m (0) = m 0 , where the system comprises a stationary Hamilton-Jacobi equation in the contact case and an evolutionary continuity equation. Then, for any fixed λ > 0, let ( u λ , m λ ) be a solution of the system H ( x , λ u λ , Du λ ) = F ( x, m λ (t)) + c ( m λ (t)), x ∈ M , ∀ t ∈ [0, T ], ∂ t m λ - div (m λ ∂H / ∂ p ( x, λu λ , Du λ ) = 0, ( x, t ) ∈ M × (0, T ], m (0) = m 0 , where c ( m λ ( t )) is the Mañé critical value of the Hamiltonian H ( x , 0, p ) — F ( x , m λ ( t )). We investigate the selection problem for the limit of ( u λ , m λ ) as λ tends to 0.