On the vanishing discount problem from the negative direction

It has been proved in [7] that the unique viscosity solution of λu_λ + H(x, dxu_λ) = 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.