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Bifurcation from bubbles in nonconvex cones

Filomena Pacella, Camilla Chiara Polvara, Luigi Provenzano

TL;DR

The work analyzes the critical exponent Neumann problem for semilinear equations in cones and shows symmetry breaking can occur in nonconvex cones when the first Neumann eigenvalue λ1(D) crosses N−1. It constructs a parametric family of spherical domains Dα via sphere diffeomorphisms, tracks λ1(Dα) through this family, and proves, under a simple eigenvalue assumption at the crossing, a Crandall–Rabinowitz bifurcation from the standard bubble yields a branch of nonradial, positive solutions. The bifurcation is shown to be global via Rabinowitz’s alternative, and the analysis hinges on a careful spectral decomposition tied to the geometry of the base domain on the sphere and the induced cone perturbations. This connects spherical-domain spectral data to nonlinear bifurcation phenomena for critical elliptic equations on cones, advancing understanding of symmetry breaking in geometric PDEs.

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 $D\subset§^{N-1}$, 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.

Bifurcation from bubbles in nonconvex cones

TL;DR

The work analyzes the critical exponent Neumann problem for semilinear equations in cones and shows symmetry breaking can occur in nonconvex cones when the first Neumann eigenvalue λ1(D) crosses N−1. It constructs a parametric family of spherical domains Dα via sphere diffeomorphisms, tracks λ1(Dα) through this family, and proves, under a simple eigenvalue assumption at the crossing, a Crandall–Rabinowitz bifurcation from the standard bubble yields a branch of nonradial, positive solutions. The bifurcation is shown to be global via Rabinowitz’s alternative, and the analysis hinges on a careful spectral decomposition tied to the geometry of the base domain on the sphere and the induced cone perturbations. This connects spherical-domain spectral data to nonlinear bifurcation phenomena for critical elliptic equations on cones, advancing understanding of symmetry breaking in geometric PDEs.

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 , 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.

Paper Structure

This paper contains 11 sections, 17 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometric preliminaries, and a class of diffeomorphisms on the sphere
  3. Neumann eigenvalues on spherical domains
  4. A family of diffeomorphisms on the sphere
  5. The induced diffeomorphisms on the cone
  6. Derivative of $\lambda_1(D_{\alpha})$ with respect to $\alpha$
  7. Proof of Proposition \ref{['limit_L2']}
  8. An example of simple eigenvalue $N-1$
  9. Bifurcation results
  10. Spectral analysis
  11. Main results

Key Result

Theorem 1.1

CPP Let $\Sigma_D$ be a cone such that $\bar{D}\subset \mathbb{S}^{N-1}_+$ and $\lambda_1(D)<N-1$, where $\mathbb{S}^{N-1}_+$ is the half-sphere. Then the minimizers of the Sobolev quotient are nonradial and hence there exist nonradial solutions $v$ of printro which are also fast decaying, i.e.

Figures (1)

  • Figure 1:

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Corollary 2.2
  • Proposition 2.3
  • proof : Proof of Proposition \ref{['3.3']}
  • Proposition 2.4
  • proof
  • Remark 2.5
  • proof : Proof of Proposition \ref{['limit_L2']}
  • ...and 25 more