Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equations

We study the asymptotic behavior of solutions of an equation of the form \begin{equation}\label{abs}\tag{*} G\big(x, D_x u,\lambda u(x)\big) = c_0\qquad\hbox{in $M$} \end{equation} on a closed Riemannian manifold $M$, where $G\in C(T^*M\times\mathbb{R})$ is convex and superlinear in the gradient variable, is globally Lipschitz but not monotone in the last argument, and $c_0$ is the critical constant associated with the Hamiltonian $H:=G(\cdot,\cdot,0)$. By assuming that $\partial_u G(\cdot,\cdot,0)$ satisfies a positivity condition of integral type on the Mather set of $H$, we prove that any equi-bounded family of solutions of \eqref{abs} uniformly converges to a distinguished critical solution $u_0$ as $\lambda \to 0^+$. We furthermore show that any other possible family of solutions uniformly diverges to $+\infty$ or $-\infty$. We then look into the linear case $G(x,p,u):=a(x)u + H(x,p)$ and prove that the family $(u_\lambda)_{\lambda \in (0,\lambda_0)}$ of maximal solutions to \eqref{abs} is well defined and equi-bounded for $\lambda_0>0$ small enough. When $a$ changes sign and enjoys a stronger localized positivity assumption, we show that equation \eqref{abs} does admit other solutions too, and that they all uniformly diverge to $-\infty$ as $\lambda \to 0^+$. This is the first time that converging and diverging families of solutions are shown to coexist in such a generality.