*Published Paper*

**Inserted:** 12 jul 2023

**Last Updated:** 12 jul 2023

**Journal:** Discrete Contin. Dyn. Syst.

**Volume:** 41

**Number:** 5

**Pages:** 2377–2389

**Year:** 2021

**Abstract:**

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.**