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 \(u_0^-\). 

　　Publication: 

　　Journal of Dynamics and Differential Equations (2022) 

　　https://doi.org/10.1007/s10884-022-10227-1 

　　Author: 

　　Ya-Nan Wang 

　　School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097, China 

　　Jun Yan 

　　School of Mathematical Sciences, Fudan University and Shanghai Key Laboratory for Contemporary Applied Mathematics, Shanghai, 200433, China 

　　Jianlu Zhang 

　　Hua Loo-Keng Key Laboratory of Mathematics & Mathematics Institute, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China 

　　Email: jzhang87@amss.ac.cn