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

Matteo Bonforte, Filippo Gazzola, Gabriele Grillo, Juan Luis Vázquez

Published 2011-04-19, updated 2011-05-01Version 2

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.