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

Elvise Berchio; Alberto Ferrero; Debdip Ganguly; Prasun Roychowdhury

doi:10.1016/j.jde.2023.03.009

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arXiv · 2211.10127

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

Elvise Berchio·Alberto Ferrero·Debdip Ganguly·Prasun Roychowdhury

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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.

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Elvise Berchio, Alberto Ferrero, Debdip Ganguly, Prasun Roychowdhury. 2022-11-18. Classification of radial solutions to $-\Delta_g u=e^u$ on Riemannian models. https://doi.org/10.1016/j.jde.2023.03.009

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