Separation of branches of O(N-1)-invariant solutions for a semilinear elliptic equation

TL;DR

This paper investigates bifurcation phenomena in semilinear elliptic equations within an annulus, introducing a novel approach to separate solution branches using O(N-1)-invariant cones, leading to insights on their unboundedness.

Contribution

It introduces two cones of O(N-1)-invariant functions that effectively separate bifurcating solution branches, demonstrating their unboundedness in a semilinear elliptic problem.

Findings

Successfully separates bifurcating solution branches using invariant cones.

Proves the unboundedness of these solution branches.

Provides a new method for analyzing symmetry-breaking bifurcations.

Abstract

We consider a semilinear elliptic problem in an annulus of R^N, with N>1. Recent results ensure that there exists a sequence p_k of exponents of the nonlinear term at which a nonradial bifurcation from the radial solution occurs. Exploiting the properties of O(N-1)-invariant spherical harmonics, we introduce two suitable cones K^1 and $K^2$ of O(N-1)-invariant functions that allow to separate the branches of bifurcating solutions from the others, getting the unboundedness of these branches.