TY - JOUR

T1 - Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

AU - Case, Jeffrey S.

AU - Moreira, Ana Claudia

AU - Wang, Yi

PY - 2019/6/1

Y1 - 2019/6/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00526-019-1566-4

DO - 10.1007/s00526-019-1566-4

M3 - Article

AN - SCOPUS:85066946375

VL - 58

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 3

M1 - 106

ER -