Classification of radial solutions to $-\Delta_g u=e^u$ on Riemannian models

TL;DR

This paper classifies radial solutions to the nonlinear PDE $- riangle_g u=e^u$ on Riemannian models, analyzing their asymptotic behavior, stability, and intersections, with a focus on how dimension influences these properties.

Contribution

It provides a comprehensive classification of solutions on Riemannian models, extending known Euclidean results to more general manifolds with various curvature bounds.

Findings

Different behaviors occur for dimensions 2-9 and 10 or higher.

Stability and intersection properties depend on the dimension.

The classification includes asymptotic behavior and stability analysis.

Abstract

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 $N\ge 2$. 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 $2\leq N\le 9$ or $N\geq 10$, respectively. The crucial role of these dimensions in classifying solutions is well-known in Euclidean space.