On the vanishing discount problem from the negative direction

It has been proved in [10] that the unique viscosity solution of \begin{equation}\label{abs}\tag{*} \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 \eqref{abs} are not unique, in general, so we focus on the asymptotics of the minimal solution $u_\lambda^-$ of \eqref{abs}. 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 \eqref{abs} admits a unique solution for $\lambda<0$ as well.