Nondegeneracy of Positive solutions of a Semilinear elliptic problem in the Hyperbolic space

TL;DR

This paper investigates the nondegeneracy of positive finite energy solutions to a semilinear elliptic equation in hyperbolic space, showing degeneracy occurs only in an N-dimensional subspace and proving nondegeneracy in geodesic balls.

Contribution

It establishes the precise conditions under which solutions are nondegenerate in hyperbolic space, highlighting the special case of geodesic balls.

Findings

Degeneracy occurs only in an N-dimensional subspace.

Positive solutions are nondegenerate in geodesic balls.

Provides a characterization of solution stability in hyperbolic space.

Abstract

In this article we will study the nondegeneracy properties of positive finite energy solutions of a semilinear elliptic equation in the N-dimensional Hyperbolic space. We will show that the degeneracy occurs only in an N dimensional subspace. We will prove that the positive solutions are nondegenerate in the case of geodesic balls.