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

Elvise Berchio; Alberto Ferrero; Debdip Ganguly; Prasun Roychowdhury

doi:10.1016/j.jde.2023.03.009

paper

Classification of radial solutions to $- Δ_{g} u = e^{u}$ on Riemannian models

arXiv:2211.10127 · doi:10.1016/j.jde.2023.03.009

Abstract

We provide a complete classification with respect to asymptotic behaviour, stability and intersections properties of radial smooth solutions to the equation $- Δ_{g} u = e^{u}$ on Riemannian model manifolds $(M, g)$ in dimension $N \geq 2$ . 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 $2 \leq N \leq 9$ or $N \geq 10$ , respectively. The crucial role of these dimensions in classifying solutions is well-known in Euclidean space.

26 pages

Classification of radial solutions to −Δg​u=eu on Riemannian models

Abstract

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Classification of radial solutions to $- Δ_{g} u = e^{u}$ on Riemannian models