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

2018/09/01 by Case, Jeffrey S., Moreira, Ana Claudia, Wang, Yi
#53A30 #58J32 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1809.00104

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 2k

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