Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space

TL;DR

This paper investigates radial solutions to the Emden-Fowler equation on hyperbolic space, revealing unique properties and asymptotic behaviors of solutions, including those with infinite energy, contrasting with Euclidean cases.

Contribution

It provides a detailed analysis of infinite energy solutions and their asymptotic behavior, extending understanding beyond finite energy solutions in hyperbolic space.

Findings

Characterization of asymptotic behavior of solutions

Distinction between finite and infinite energy solutions

Identification of unique properties in hyperbolic setting

Abstract

We study the Emden-Fowler equation $-\Delta u=|u|^{p-1}u$ on the hyperbolic space ${\mathbb H}^n$. We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is $p=(n+2)/(n-2)$ as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers \cite{mancini, bhakta} consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy solutions.