Bifurcation from bubbles in nonconvex cones

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.