Existence and Nonexistence for Hessian Exterior Dirichlet Problems with \(k\)-Admissible Asymptotic Matrices

We study exterior Dirichlet problems for $k$-Hessian equations with prescribed quadratic asymptotics, allowing the asymptotic matrix to be merely $k$-admissible and not necessarily positive definite. The key point is that the correct metric at infinity is not determined by the asymptotic matrix itself, but by the coefficient matrix obtained by linearizing the $k$-Hessian operator at this matrix. This gives the exterior barriers and subsolutions needed to solve the Dirichlet problem, both in viscosity and smooth settings, for all sufficiently large asymptotic constants. In the case of smooth, strictly star-shaped domains with strictly $(k-1)$-convex boundary, we complete the characterization of existence and nonexistence through a linearized capacitary comparison and a tangential-trace contradiction on the inner boundary.