Classification of radial solutions to −Δu = e on Riemannian models

We provide a complete classification with respect to asymptotic behaviour, stability and intersections properties of radial smooth solutions to the equation $-\Delta_g u=e^u$ on Riemannian model manifolds (M,g) in dimension N>1. 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 $2\leq N \leq 9$ or $N\geq 10$, respectively. The crucial role of these dimensions in classifying solutions is well-known in Euclidean space; here the analysis highlights new properties of solutions that cannot be observed in the flat case.