In this paper, we deal with an overdetermined problem for the k-Hessian equation (1 k< n2) in the exterior domain and prove the corresponding ball characterizations. Since that Weinberger type approach seems to fail to solve the problem, we give a new perspective to solve exterior overdetermined problem by combining two integral identities and geometric inequalities inspired by Brandolini-Nitsch-Salani's results BNS. Meanwhile, we establish general monotone formulas to derive geometric inequalities related to k-admissible solution u in RⁿΩ, where Ω is smooth, k-convex and star-shaped domain, which constructed by Ma-ZhangMZ and Xiaoxiao.
Yin et al. (2025) studied this question.