Convergence of solutions for some degenerate discounted Hamilton–Jacobi equations

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