On bifurcation for semilinear elliptic Dirichlet problems and the Morse-Smale index theorem

TL;DR

This paper investigates bifurcation phenomena in semilinear elliptic Dirichlet problems on star-shaped domains, introducing a bifurcation parameter via domain shrinking, and proves a special case of Smale's index theorem.

Contribution

It establishes a bifurcation result for elliptic problems with a novel approach involving domain shrinking and connects it to a special case of Smale's index theorem.

Findings

Bifurcation occurs when shrinking the domain crosses a critical size.

A new proof of a special case of Smale's index theorem is provided.

The results apply to star-shaped domains with Dirichlet boundary conditions.

Abstract

We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on star-shaped domains, where the bifurcation parameter is introduced by shrinking the domain. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.