  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 $\sigma_k$-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 $2k<\dim X$. Our examples are all such that the
boundary with its induced metric is a Riemannian product of a round sphere with
an Einstein manifold.
