Vanishing discount limits for 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, λ u) = 0 H ( x , D u , λ u ) = 0 in Rⁿ R n as λ → 0⁺ λ → 0 + . Assuming that the Aubry set is localized, we use 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.