### Abstract

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing σ_{k}-curvature in the interior and constant H_{k}-curvature on the boundary. When restricting to the closure of the positive k-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem which allows us to construct examples of compact Riemannian manifolds (X, g) for which this problem admits multiple non-homothetic solutions in the case when 2 k< dim X. Our examples are all such that the boundary with its induced metric is a Riemannian product of a round sphere with an Einstein manifold.

Original language | English (US) |
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Article number | 106 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 58 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jun 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Calculus of Variations and Partial Differential Equations*,

*58*(3), [106]. https://doi.org/10.1007/s00526-019-1566-4