Stability and qualitative properties of radial solutions of the Lane-Emden-Fowler equation on Riemannian models

TL;DR

This paper investigates the existence, uniqueness, and stability of radial solutions to the Lane-Emden-Fowler equation on various Riemannian models, including hyperbolic space, highlighting how curvature affects solution properties.

Contribution

It extends analysis of the Lane-Emden-Fowler equation to Riemannian models with diverse curvature, exploring stability and asymptotic behavior of solutions.

Findings

Sign properties and asymptotic behavior depend on the critical Sobolev exponent.

Stability is influenced by the Joseph-Lundgren exponent.

Results include models with unbounded below sectional curvatures.

Abstract

We study existence, uniqueness and stability of radial solutions of the Lane-Emden-Fowler equation $-\Delta_g u=|u|^{p-1}u$ in a class of Riemannian models $(M,g)$ of dimension $n\ge 3$ which includes the classical hyperbolic space $\mathbb H^n$ as well as manifolds with sectional curvatures unbounded below. Sign properties and asymptotic behavior of solutions are influenced by the critical Sobolev exponent while the so-called Joseph-Lundgren exponent is involved in the stability of solutions.