*Published Paper*

**Inserted:** 13 jul 2023

**Last Updated:** 13 jul 2023

**Journal:** Comm. Partial Differential Equations

**Volume:** 48

**Number:** 4

**Pages:** 576–622

**Year:** 2023

**Abstract:**

We study the asymptotic behavior of the viscosity solutions $u^\lambda_G$ of the discounted Hamilton-Jacobi (HJ) equation \[ \lambda u(x)+G(x,u')=c(G)\qquad\hbox{in $M$} \] as the positive discount factor $\lambda$ tends to 0, where $G(x,p):=H(x,p)-V(x)$ is the perturbation of a Hamiltonian $H\in\hbox{C}(TM)$, $\mathbb{Z}$--periodic in the space variable and convex and coercive in the momentum, by a compactly supported potential $V\in\hbox{C}_c(\mathbb R)$. The constant $c(G)$ appearing above is defined as the infimum of values $a\in\mathbb R$ for which the HJ equation $G(x,u')=a$ in $M$ admits bounded viscosity subsolutions. We prove that the functions $u^\lambda_G$ locally uniformly converge, for $\lambda\rightarrow 0^+$, to a specific solution $u_G^0$ of the critical equation \[ G(x,u')=c(G)\qquad\hbox{in $M$}. \] We identify $u^0_G$ in terms of projected Mather measures for $G$ and of the limit $u^0_H$ to the unperturbed periodic problem. Our work also includes a qualitative analysis of the critical equation with a weak KAM theoretic flavor.