On the vanishing discount problem from the negative direction

It has been proved in [7] that the unique viscosity solution of

$ \begin{equation} \lambda u_\lambda+H(x,d_x u_\lambda) = c(H)\qquad\hbox{in $M$}, \;\;\;\;\;\;\;\;\;(*)\end{equation} $

uniformly converges, for $ \lambda\rightarrow 0^+ $, to a specific solution $ u_0 $ of the critical equation

$ H(x,d_x u) = c(H)\qquad\hbox{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 $ \lambda\rightarrow 0^- $. In this case, viscosity solutions of equation (*) are not unique, in general, so we focus on the asymptotics of the minimal solution $ u_\lambda^- $ of (*). Under the assumption that constant functions are subsolutions of the critical equation, we prove that the $ u_\lambda^- $ also converges to $ u_0 $ as $ \lambda\rightarrow 0^- $. Furthermore, we exhibit an example of $ H $ for which equation (*) admits a unique solution for $ \lambda<0 $ as well.