Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

Jeffrey S. Case, Ana Claudia Moreira, Yi Wang

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 σk-curvature in the interior and constant Hk-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 languageEnglish (US)
Article number106
JournalCalculus of Variations and Partial Differential Equations
Volume58
Issue number3
DOIs
StatePublished - Jun 1 2019

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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