Convergence of the solutions of the discounted equation

We consider a continuous coercive Hamiltonian $H$ on the cotangent bundle of the compact connected manifold $M$ which is convex in the momentum. If $u_{\lambda }:M\to \mathbb{R}$ is the viscosity solution of the discounted equation $$\lambda u_{\lambda }(x)+H(x,d_x{u}_{\lambda })=c(H),$$ where $c(H)$ is the critical value, we prove that $u_{\lambda }$ converges uniformly, as $\lambda \to 0$, to a specific solution $u_0:M\to \mathbb{R}$ of the critical equation $$H(x,d_{x}u)=c(H)\mathrm{.}$$ We characterize $u_0$ in terms of Peierls barrier and projected Mather measures.