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, Du) = F (x, m(t)), x ∈ M, ∀ t ∈ [0, T], ∂tm - div (m ∂H/∂p(x, u, Du) = 0, (x, t) ∈ M × (0, T], m(0) = m0, 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 (uλ,mλ) be a solution of the system H(x, λuλ, Duλ) = F (x, mλ(t)) + c(mλ(t)), x ∈ M, ∀t ∈ [0, T], ∂tmλ - div (mλ∂H/∂p(x, λuλ, Duλ) = 0, (x, t) ∈ M × (0, T], m(0) = m0, where c(mλ(t)) is the Mañé critical value of the Hamiltonian H(x, 0,p) — F(x,mλ(t)). We investigate the selection problem for the limit of (uλ, mλ) as λ tends to 0.