Classification of radial solutions to $-{\mathrm{\Delta }}_{g}u=e^u$ on Riemannian models

We provide a complete classification with respect to asymptotic behaviour, stability and intersections properties of radial smooth solutions to the equation $-{\mathrm{\Delta }}_{g}u=e^u$ on Riemannian model manifolds $(M,g)$ in dimension $N\ge 2$. Our assumptions include Riemannian manifolds with sectional curvatures bounded or unbounded from below. Intersection and stability properties of radial solutions are influenced by the dimension $N$ in the sense that two different kinds of behaviour occur when $2\le N\le 9$ or $N\ge 10$, respectively. The crucial role of these dimensions in classifying solutions is well-known in Euclidean space.