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 {equation*} \{{aligned} &H(x,u,Du)=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, {aligned}. {equation*} 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 {equation*} \{ {aligned} &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,λ u^λ,Du^λ))=0, &&(x,t)∈ M×(0,T],\\ &m(0)=m_0, {aligned}. {equation*} where c(m^λ(t)) is the Ma\~n\'e 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.