  We study a class of elliptic problems, involving a $k$-Hessian and a very
fast-growing nonlinearity, on a unit ball. We prove the existence of a radial
singular solution and obtain its exact asymptotic behavior in a neighborhood of
the origin. Furthermore, we study the multiplicity of regular solutions and
bifurcation diagrams. An essential ingredient of this study is analyzing the
number of intersection points between the singular and regular solutions for
rescaled problems. In the particular case of the exponential nonlinearity, we
obtain the convergence of regular solutions to the singular and analyze the
intersection number depending on the parameter $k$ and the dimension $d$.
