On the vanishing discount problem from the negative direction

It has been proved in [10] that the unique viscosity solution of λuλ +H(x,dxuλ)=c(H) in M (*) uniformly converges, for λ → 0+, to a specific solution u0 of the critical equation H(x,dxu) = c(H) in M, where M is a closed and connected Riemannian manifold and c(H) is the critical value. In this note, we consider the same problem for λ → 0−. In this case, viscosity solutions of equation (*) are not unique, in general, so we focus on the asymptotics of the minimal solution u−λ of (*). Under the assumption that constant functions are subsolutions of the critical equation, we prove that the u−λ also converges to u0 as λ → 0−. Furthermore, we exhibit an example of H for which equation (*) admits a unique solution for λ < 0 as well.