Convergence of solutions for some degenerate discounted Hamilton--Jacobi equations

We study solutions of Hamilton–Jacobi equations of the form $$\lambda \alpha (x)u_{\lambda }(x)+H(x,D_x{u}_{\lambda })=c,$$ where $\alpha$ is a nonnegative function, $\lambda$ a positive constant, $c$ a constant and $H$ a convex coercive Hamiltonian. Under suitable conditions on $\alpha$ we prove that the functions $u_{\lambda }$ converge as $\lambda \to 0$ to a function $u_0$ that is a solution of the critical equation $H(x, D_x u_0) = c$.