On the negative limit of viscosity solutions for discounted Hamilton-Jacobi equations

Suppose $M$ is a closed Riemannian manifold. For a $C^2$ generic (in the sense of Ma\~n\'e) Tonelli Hamiltonian $H: T^*M\rightarrow\mathbb{R}$, the minimal viscosity solution $u_\lambda^-:M\rightarrow \mathbb{R}$ of the negative discounted equation \[-\lambda u+H(x,d_xu)=c(H),\quad x\in M,\ \lambda>0 \] with the Ma\~n\'e's critical value $c(H)$ converges to a uniquely established viscosity solution $u_0^-$ of the critical Hamilton-Jacobi equation \[ H(x,d_x u)=c(H),\quad x\in M \] as $\lambda\rightarrow 0_+$. We also propose a dynamical interpretation of $u_0^-$.