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

By Matteo Bonforte, Filippo Gazzola, G. Grillo, Juan Luis V'azquez. Calculus of Variations and Partial Differential Equations, 2011

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We study the Emden–Fowler equation −Δu = |u|p−1u on the hyperbolic space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb H}^n}$$\end{document} . 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 (Bhakta and Sandeep, Poincaré Sobolev equations in the hyperbolic space, 2011; Mancini and Sandeep, Ann Sci Norm Sup Pisa Cl Sci 7(5):635–671, 2008) 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.

Subfields: Applied Mathematics, Mathematical Physics

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