Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains

We study the asymptotic behavior of solutions to the fully nonlinear Hamilton-Jacobi equation $H(x,Du,\lambda u)=0$ in ${\mathbb{R}}^n$ as $\lambda \to 0^{+}$. Under the assumption that the Aubry set is localized, we employ a variational approach to derive limiting Mather-type measures and formulate a selection principle. Central to our analysis is a modified variational formula that bridges global and local state-constraint solutions, thereby extending localization techniques to the nonlinear framework.