On the subcritical Lane-Emden equation on Riemannian models with polynomial volume growth

TL;DR

This paper investigates the existence and non-existence of positive solutions to the subcritical Lane-Emden equation on Riemannian models with polynomial volume growth, revealing complex phenomena depending on the subcritical regime.

Contribution

It classifies the subcritical regimes into three types and demonstrates the possibility of multiple solutions in the intermediate regime on certain Riemannian models.

Findings

Existence in slightly subcritical regime

Non-existence in strongly subcritical regime

Multiple solutions in the intermediate regime

Abstract

We focus on the problems of existence and non-existence of positive solutions for the Sobolev-subcritical Lane-Emden equation on certain Riemannian manifolds (mainly models) with asymptotically negative curvature, which, from the viewpoint of the volume growth of geodesic balls, can be regarded as intermediate settings between the Euclidean and the hyperbolic spaces. A number of interesting phenomena arise: the subcritical regime naturally divides into three further ranges, characterized by existence phenomena (slightly subcritical), non-existence phenomena (strongly subcritical), and by a mixed behavior where existence and non-existence strongly depend on additional assumptions on the manifold (intermediate). In the intermediate regime, we further show that the radial homogeneous Dirichlet problem in geodesics balls may admit multiple positive solutions, thereby revealing substantial differences with respect to both the Euclidean and the hyperbolic settings.