Structure of radial solutions to a Hénon type equation with exponential nonlinearity on the hyperbolic space

In this paper, we study the separation and stability properties of radial solutions to a Hénon-type equation with exponential nonlinearity on the hyperbolic space. By transforming the radial hyperbolic equation into a weighted Euclidean equation and applying well-established separation results in Euclidean space, we classify the solution structures or establish sharp alternatives for radial solutions throughout the full range $n\ge 2$ and $\alpha >-2$. We also give an affirmative answer to the open question recently posed by Huang and Zhao.