In this paper, we prove the existence of a solution for the exterior Dirichlet problem for Hessian equations on a non-convex ring. Moreover, the solution we obtained is smooth. This extends the result of Bao-Li-Li, ``On the exterior Dirichlet problem for Hessian equations'' Trans. Amer. Math. Soc.366(2014).
Li et al. (Fri,) studied this question.