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