We consider $sigma_k$-curvature equation with $H_k$-curvature condition on a compact manifold with boundary $(X^{n+1}, M^n, g)$. When  restricting to the closure of the positive $k$-cone, this is a fully nonlinear elliptic equation with a fully nonlinear Robin-type boundary  condition. We prove a general bifurcation theorem in order to study nonuniqueness of solutions when $2k<n+1$. We explicitly give examples of  product manifolds with multiple solutions. It is analogous to Schoen’s example for Yamabe problem on $S^1times S^{n-1}$. This is joint work  with Jeffrey Case and Ana Claudia Moreira.