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 $$H(x,d_x u)=c(H).$$ We characterize $u_0$ in terms of Peierls barrier and projected Mather measures.