On bifurcation for semilinear elliptic Dirichlet problems on geodesic balls

TL;DR

This paper investigates bifurcation phenomena in semilinear elliptic Dirichlet problems on geodesic balls, using the radius as a bifurcation parameter, and establishes a special case of Smale's index theorem.

Contribution

It introduces a bifurcation analysis on geodesic balls and proves a specific case of Smale's index theorem within this context.

Findings

Identification of bifurcation points as the radius varies

Derivation of a special case of Smale's index theorem

Insights into the structure of solutions for semilinear elliptic problems

Abstract

We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.