Bifurcation from bubbles in nonconvex cones

TL;DR

This paper studies how symmetry breaking occurs in solutions to a critical elliptic equation in nonconvex cones, using bifurcation theory to identify when nonradial solutions emerge from radial ones.

Contribution

It establishes the existence of a bifurcation of nonradial solutions from the standard bubble in nonconvex cones, linking it to eigenvalues of the Laplace-Beltrami operator.

Findings

Bifurcation occurs when the first eigenvalue crosses a critical threshold.

Existence of a global bifurcation branch of nonradial solutions.

Symmetry breaking is related to the spectral properties of the domain.

Abstract

We investigate the Neumann problem for the critical semilinear elliptic equation in cones. The standard bubble provides a family of radial solutions, which are known to be the only positive solutions in convex cones. For nonconvex cones, symmetry breaking may occur and the symmetry breaking is related to the first nonzero Neumann eigenvalue of the Laplace Beltrami operator on the domain $D\subset\S^{N-1}$, that spans the cone. We construct a one-parameter family of domains on the sphere whose first eigenvalue crosses the threshold at which the bubble loses stability. Under the assumption that this eigenvalue is simple, we prove, via the Crandall Rabinowitz bifurcation theorem, the existence of a branch of nonradial solutions bifurcating from the standard bubble. Moreover we show that the bifurcation is global.