  For a continuous Hamiltonian $H : (x, p, u) \in T^*\mathbb{R}^n \times
\mathbb{R}\rightarrow \mathbb{R}$, we consider the asymptotic behavior of
associated Hamilton--Jacobi equations with state-constraint $H(x, Du, \lambda
u) \leq C_\lambda$ in $\Omega_\lambda\subset \mathbb{R}^n$ and $H(x, Du,
\lambda u) \geq C_\lambda$ on $\overline{\Omega}_\lambda\subset \mathbb{R}^n$ a
$\lambda\rightarrow 0^+$. When $H$ satisfies certain convex, coercive, and
monotone conditions, the domain $\Omega_\lambda:=(1+r(\lambda))\Omega$ keeps
bounded, star-shaped for all $\lambda>0$ with $\lim_{\lambda\rightarrow
0^+}r(\lambda)=0$, and $\lim_{\lambda\rightarrow 0^+}C_\lambda=c(H)$ equals the
ergodic constant of $H(\cdot,\cdot,0)$, we prove the convergence of solutions
$u_\lambda$ to a specific solution of the critical equation $H(x, Du, 0)\leq
c(H) $ in $\Omega$ and $H(x, Du, 0)\geq c(H) $ on $\overline{\Omega}$. We also
discuss the generalization of such a convergence for equations with more
  general $C_\lambda$ and $\Omega_\lambda$.
